Line a 20/20 cm (8×8 inches) baking tray with parchment paper. Preheat the oven to 180 °C (356 °F).: You will notice the kitchen begin to warm as the oven comes up to temperature, a faint dry heat that promises even baking. Getting the tray ready with parchment prevents sticking and helps you lift the whole slab out for clean slicing. Use an oven thermometer if your oven runs hot or cold, because incorrect temperature is a leading cause of overbaked or underbaked centers. If the tray is not lined securely, the batter may bake unevenly along the edges, so tuck the paper into the corners firmly.
In a thick-bottomed saucepan, melt the butter over low heat and add the broken chocolate pieces and sugar. When the mixture is homogeneous, turn off the heat.: As the butter softens and chocolate glosses, you will smell a deep, toasty cocoa aroma. Stir slowly until the mixture is smooth and shiny, then remove from heat to avoid overheating. This gentle melt preserves the chocolate temper and creates a silkier batter. If the mixture begins to steam heavily or smells burnt, the heat is too high and you should remove it immediately to prevent grainy texture.
Add the egg yolks, one at a time, and mix well.: After the chocolate cools slightly, the yolks add a rich, velvety texture and glossy shine. Add them one by one so they incorporate smoothly without scrambling from residual heat. The batter should feel glossy and cohesive after each addition. A common mistake is adding cold yolks to very hot chocolate, which can curdle the mixture, so allow the chocolate to cool for a minute first.
Separately combine the flour and cocoa powder. Add them to the chocolate cream, stirring gently.: When you fold in the dry ingredients you will notice the batter thicken and the chocolate deepen in color. Mixing gently preserves air and keeps the final crumb dense yet tender. Use a sieve to avoid lumps and fold with a spatula until just combined. Overmixing activates gluten and can turn fudgy brownies into a cakier texture, so stop as soon as streaks vanish.
Whisk the egg whites well with a pinch of salt until they turn into a nice meringue. Incorporate them into the chocolate cream with upward movements. Place the whole mixture in the pan.: Whipping the whites adds lift and a delicate crumb. You will hear a change in sound as the mixer transitions to soft peaks, and the whites will look glossy. Folding them in with gentle upward motions keeps air intact while blending them evenly into the batter. If you fold aggressively you will deflate the whites and lose that lightness, so use patient strokes and stop when the batter is uniform.
Mix the cream cheese, egg, flour, and sugar until it becomes a uniform cream. Now add the vanilla and mix.: The filling should feel smooth and slightly dense, with a mild tang and sweet edge. Softened cream cheese blends best, producing a velvety texture that bakes into a custard like swirl. Adding vanilla brightens the flavor, making it pop against the dark base. If the cream cheese is too cold you may get lumps, so let it sit at room temperature before mixing.
Place dollops of the mixture on top of the brownie. Smooth it out with a knife with a sharp tip, creating a pattern when combined with the dark mixture.: As you dot and drag the filling you will see marbled patterns form, which gives the final bars a bakery style look. The contrast between matte chocolate and creamy white is visually appealing. Use a sharp knife to create sweeping motions, cleaning the blade between moves for crisp lines. If you overwork the surface the two mixtures may blend too much and lose distinct patterning.
Wash the strawberries and remove their leaves. Place them on top of the cream cheese, one by one.: Fresh strawberries should be firm and fragrant, adding a bright scent as you arrange them. Press each berry gently into the filling so it adheres but does not sink entirely. The fruit will roast slightly in the oven, concentrating its sweetness. If the berries are too soft they can release too much juice, making the top soggy, so pick ones with a little resistance.
Bake in the oven for 25-30 minutes, until the top is cooked through and the inside is still slightly soft. Remove and leave to cool in the pan. After about 2 hours (if you can be that patient), slice it up. Serve as it is or with generous portions of ice cream.: During baking the aroma of chocolate will deepen and the cream cheese will set with a faint jiggle in the center. Watch for a set edge and a slightly soft middle, which indicates fudgy doneness. Cooling in the pan allows the structure to firm up, making cleaner slices. A common problem is overbaking, which dries the interior; remove a few minutes earlier if your oven runs hot and let residual heat finish the set.