Preheat the oven to 400 degrees F.: When you slide the rack in, the oven should already be hot so the beets begin to roast immediately, producing a sweet steam and a caramelized aroma. You may hear gentle bubbling from the roasting pan as water steams, and you should see a fragrant, slightly toasty scent fill the kitchen. Doing this sets a precise starting point for the roast time, which ensures even tenderness. A common mistake is preheating too late, which can lead to undercooked beets and a watery puree. If your oven runs cool, allow an extra five to ten minutes for roasting at the same temperature to reach the right doneness.
Wash beets and place in a small roasting pan with 1/2 cup water. Cover and bake for 45 minutes to 1 hour, or until easily pierced with a fork. Dunk the beets in cool water, then using your fingers, slip off the peels. Coarsely chop, then transfer to a blender or food processor and puree. Measure out 2 cups and set aside.: As the beets roast, you will notice an earthy steam and a subtle sweet smell developing, signaling caramelization. The skins should separate easily when cooled briefly; if they cling, they likely need more oven time. After pureeing, the texture should be velvety and smooth without large fibrous pieces. The puree should be slightly glossy, not grainy. The reason we measure exactly 2 cups is to maintain batter hydration and texture. If your puree seems watery, drain briefly to avoid thinning the batter too much. A frequent pitfall is skipping the cooling step which makes peeling messy and may lead to lost juices; let them cool just enough to handle.
Reduce the oven temperature to 375 degrees F. Lightly coat a 10-cup Bundt or tube pan with oil and dust with flour. In a large bowl, sift together the all-purpose flour, whole-wheat pastry flour, baking soda, and salt. Set aside.: Lowering the temperature prevents the cake from overbrowning while ensuring even internal baking. When you oil and flour the pan, the surface should look barely glossy with a light dusting of flour to prevent sticking and encourage an easy release. Sifting the flours and leavening distributes the baking soda evenly so the cake rises uniformly; you should see a pale, airy mixture after sifting. One mistake is skipping the flour dusting on the pan which can make unmolding the Bundt tricky, tearing the cake surface. Keep the prepared dry mix close at hand to add to the wet ingredients without delay.
In a double boiler or a small pan over low heat, add the chocolate and 1/4 cup of the oil. Heat just until the chocolate melts. Remove from the heat and stir until well combined. Set aside.: Gently melting the unsweetened chocolate with oil creates a glossy, pourable chocolate mixture with a deep scent of cocoa. You should stir until you see a smooth shine and no lumps, and the mixture should coat the back of a spoon. Heating too quickly can cause the chocolate to seize and become grainy. Using a double boiler or a very low heat keeps the chocolate temperate and silky. If it thickens as it cools, stir in a teaspoon more oil to loosen it. Avoid letting steam or water touch the chocolate which can cause it to become clumpy.
In a large mixing bowl, beat together the eggs and sugar until fluffy. Slowly add the remaining 1/4 cup oil, Greek yogurt, vanilla extract, the chocolate mixture, and the 2 cups beet puree.: As you beat the eggs and sugar , the mixture will lighten and become pale and billowy, which helps incorporate air for lift. Adding the oil and Greek yogurt slowly maintains the emulsion, so the batter stays cohesive and not separated. When you fold in the warm chocolate and the glossy beet puree, the batter will turn a rich, deep color and smell of cocoa with an earthy undertone. If you add the liquids too quickly, the mixture can break or become grainy, so pour slowly and mix gently. Watch the texture; it should be smooth and unified, not curdled. If it seems separated, a few more gentle beats will usually bring it back together.
Gently add the flour mixture, stirring by hand just until combined.: At this step you will feel the batter thicken and become more cake like, with a weightier but still soft texture. Use a rubber spatula and fold with slow, deliberate turns to avoid knocking out air. The right consistency is a ribbon that falls slowly from the spatula, and you should still see a glossy sheen from the chocolate and beet puree. Overmixing develops gluten and yields a denser result, so stop as soon as streaks of flour disappear. A common misstep is vigorous mixing, which creates a tougher crumb, so embrace a light touch here.
Pour the batter into the prepared pan.: As you pour, the batter should move slowly and settle evenly into the pan, filling grooves and creating a smooth surface. Tap the pan gently on the counter to release large air pockets; you may hear a soft thud. The batter will look thick but spreadable, and it should not be runny. If bubbles rise to the top, gently pop them with a toothpick to prevent tunnels. A mistake is overfilling the pan; leave a bit of headspace so the cake can rise without spilling over.
Bake for 45 minutes, or until a toothpick inserted near the center comes out clean. Transfer the pan to a wire rack to cool for 30 minutes. Run a thin knife along the outer and inner edges of the pan and carefully invert the cake onto the wire rack. Let it cool completely before glazing.: During baking the kitchen fills with a toasty chocolate aroma and the cake surface will set with a slight spring when touched. The edges may pull away slightly from the pan, which is a good sign. After the initial cooling, running a knife around the pan ensures the cake releases cleanly. The cake should be fully cool before glazing so the glaze does not melt into puddles; patience pays off here. A common error is unmolding too soon, which can cause the cake to break, so allow the thirty minute rest before inverting, and then wait until completely cool before adding the glaze.
For the glaze: Beat together the cream cheese and powdered sugar until smooth and light. Beat in the milk and vanilla extract. If desired, add a bit more milk or powdered sugar as needed to reach your desired glaze consistency. Drizzle over the cake, letting it run over the top and sides.: Beat together the cream cheese and powdered sugar until smooth and light. Beat in the milk and vanilla extract. If desired, add a bit more milk or powdered sugar as needed to reach your desired glaze consistency. Drizzle over the cake, letting it run over the top and sides. : The glaze should be silky, slightly tangy, and glossy, with a fragrance of vanilla that complements the deep chocolate . As you beat the softened cream cheese and sifted powdered sugar, the mixture becomes airy and smooth, then the milk loosens it into a pourable glaze. Start with the smaller amount of milk and add sparingly so you maintain control of the drip. When drizzling, use a spoon or small spatula to guide the glaze for pretty, natural runs down the sides. If the glaze is too thin, add a little more powdered sugar; if too thick, add milk a drop at a time. One mistake is glazing a warm cake which causes the glaze to melt and become runny, so ensure the cake is completely cool before decorating.