Preheat the oven to 350°F (175°C). Lightly butter an 8-inch square baking pan, then line the bottom with parchment or wax paper. Place the pan on a baking sheet for stability during baking.

In a small bowl, mix 2 tablespoons of the sugar with 2 1/2 teaspoons of ground cinnamon and the espresso powder. Set aside.

In a large bowl, whisk together the flour, remaining 1 1/4 cups of sugar, remaining 1 tablespoon of cinnamon, baking powder, and salt.

In a separate bowl, whisk the milk, eggs, and vanilla extract until fully combined. Pour the wet ingredients into the dry mixture and gently whisk until a smooth batter forms.

Using a rubber spatula, carefully fold in the cooled, melted butter until the batter is smooth and glossy. Do not overmix.

Pour half of the batter into the prepared baking pan and spread evenly. Sprinkle the chocolate chips and cinnamon chips over the batter, followed by an even layer of the prepared cinnamon sugar-espresso mixture. Pour the remaining batter over the top and smooth the surface.

Place the cake in the oven and bake for 35 to 40 minutes, or until it has risen and slightly pulled away from the sides of the pan. A thin knife inserted in the center should come out clean. Remove the cake from the oven and allow it to rest in the pan for 15 minutes.

Carefully unmold the cake onto a wire rack. Peel off the parchment or wax paper, then invert the cake back onto another rack to cool completely with the top facing up.

In a heatproof bowl set over a saucepan of gently simmering water, melt the chopped bittersweet chocolate and butter. Stir continuously until the mixture is smooth, thick, and glossy. Do not overheat.

Once the cake is fully cooled, spread the frosting evenly over the surface using an offset spatula. Allow the frosting to set at room temperature before slicing and serving.