In a medium bowl, mix together the melted butter and sugar until combined. Whisk in the vanilla and melted chocolate. Add the egg and whisk until incorporated.: When you stir this mixture you'll notice a glossy consistency and a warm chocolate aroma, signaling the melted chocolate has blended with the sugars. I always let the melted butter and chocolate cool until just warm so the egg folds in without cooking, and whisk until the mixture feels smooth under the whisk, not grainy. If the mixture looks separated, keep whisking gently and let it rest for a few minutes so temperatures equalize. A common mistake is adding the egg to hot butter which can create scrambled bits, so double check temperature by touching the bowl. The sound is quiet, but you may hear a slight glug as air incorporates, and the color deepens to a uniform chocolate brown.
In another bowl, whisk together flour, cocoa, salt, and baking soda. Stir flour mixture into the chocolate mixture, using your hands if needed to form a smooth dough. Divide the dough in two, and either roll each piece to 1/4-inch thick between two pieces of waxed paper, or roll into two 2 1/2-inch diameter logs. Refrigerate at least two hours or until firm.: As you whisk the dry elements, the cocoa will puff and the combined aroma becomes earthy. When you add these to the wet, expect the batter to thicken rapidly; the texture should change from glossy to pliable and slightly tacky under your fingers. Press a piece between your palms to check cohesion, it should hold without crumbling. Rolling between waxed paper yields an even thickness and fewer fingerprints, while logs are convenient for slicing; whichever you choose, chill time firms the fat crystals and prevents spreading during baking. Overworking here will yield a tough cookie, so handle with gentle, decisive motions. The dough chilling is crucial, and if it remains sticky after two hours, pop it back in until firm to the touch.
Heat oven to 325 degrees and line baking sheet with parchment paper or a silicone baking mat.: The oven set to 325 degrees produces a gentle bake that allows the edges to set while the centers remain tender, minimizing cracked shells. Lining ensures easy release and even browning; you should hear a faint hiss of settling heat when you slide the tray in. If the oven has hot spots, rotate the tray halfway through to keep color consistent. A common pitfall is using a dark pan which accelerates browning, so prefer light colored pans or use an extra minute less baking time and watch the edges. The smell before baking is mostly neutral, but once in the oven you will begin to detect toasty cocoa notes.
Cut dough logs into quarter inch thick slices, or cut out 2 1/2-inch diameter circles from dough sheets. Arrange on prepared baking sheet, leaving at least 1 inch between cookies. Bake for 15 minutes or until cookies are set and just starting to darken around the edges.: The slices should hold their shape as you place them, and the spacing allows air to circulate so each cookie bakes evenly. In the oven you will observe the edges firming first and a faint rise before settling, and the smell becomes deeper, almost brownie like. The surface will lose its glossy sheen and feel slightly dry to the touch when done. Avoid overbaking which will make cookies crisp rather than tender; remove them when the centers still look slightly soft because carryover cooking will finish them. A typical mistake is crowding the tray, which leads to uneven baking and merged cookies.
To prepare filling, beat butter and brown sugar in a large bowl with an electric mixer on medium speed until light and fluffy. Mix in flour, salt, and vanilla extract on low speed until incorporated. Add cream and beat until fluffy, about 2 minutes, adding more cream if necessary to achieve desired consistency. Stir in chocolate chips. Spread or pipe about 1 tablespoon of filling onto underside of half of the cooled cookies, then top with remaining cookies and press lightly to adhere.: As you cream the butter and brown sugar, you will notice the mixture lighten in color and develop air pockets, which is the visual cue for fluffiness. Adding the flour on low prevents dusting and keeps the filling smooth rather than grainy. The cream adjusts texture, and the filling should be creamy and hold soft peaks before adding chips. When folding in mini semisweet chocolate chips , you will see little specks throughout and taste bursts of chocolate in each bite. Pipe or spread the filling evenly, feeling for slight resistance that tells you there is enough body to hold the cookie sandwich together. A trap to avoid is over beating after adding the liquid which can thin the filling excessively, so pause frequently to check consistency. After assembling, press just enough to see the filling spread to the edges, ensuring each bite has balance of shell and center.
Cookies will keep, refrigerated in an airtight container, for up to one week; let come to room temperature before eating.: Proper storage maintains texture and flavor, and refrigeration keeps the filling firm and safe. When you remove them from the fridge, allow 20 to 30 minutes at room temperature so the filling softens and flavors bloom; the aroma becomes more pronounced and the texture more tender. If you notice condensation forming, let them sit briefly uncovered on a rack before sealing to avoid sogginess. A common error is leaving them at room temperature for long stretches, which can shorten shelf life, so keep them chilled if not eaten within a day or two.