Preheat the oven to 350°F. Butter an 8-inch square baking pan and line the bottom with parchment paper or wax paper. Place the prepared pan on a baking sheet.
In a small bowl, combine 2 tbsp sugar, 2 1/2 tsp ground cinnamon, and the instant espresso powder. Stir until evenly blended, then set aside.
In a large mixing bowl, whisk together the all-purpose flour, the remaining 1 1/4 c sugar, baking powder, salt, and the remaining 1 tbsp ground cinnamon.
In a separate small bowl, whisk together the whole milk, eggs, and vanilla extract until fully combined.
Pour the milk mixture over the flour mixture. Gently whisk until a smooth, uniform batter forms.
Using a rubber spatula, fold in the melted and cooled butter until it is fully absorbed into the batter. The batter should appear smooth and glossy.
Spread half of the batter evenly into the prepared baking pan.
Sprinkle the mini chocolate chips and cinnamon chips evenly over the batter. Dust the cinnamon-sugar espresso mixture over the chips.
Spread the remaining batter over the filling layer and smooth the top.
Bake for 35 to 40 minutes, or until the cake is puffed, beginning to pull away from the sides of the pan, and a thin knife inserted into the center comes out clean.
Transfer the cake to a cooling rack and allow it to rest for 15 minutes.
Carefully unmold the cake onto another rack. Remove the parchment or wax paper, then invert the cake back onto the first rack so it is right side up. Allow the cake to cool completely to room temperature.
To prepare the frosting, place the finely chopped bittersweet chocolate and butter in a heatproof bowl.
Set the bowl over a saucepan of gently simmering water. Stir frequently until the chocolate and butter are melted and smooth.
Do not overheat the frosting. It should remain smooth, glossy, thick, and spreadable. If the frosting becomes too thin, allow it to stand at room temperature until it thickens slightly.
Using an offset spatula, spread the frosting over the cooled cake in even, generous swirls.
Allow the frosting to set at room temperature before slicing and serving.