For the Cake:
1¼ cups plus 2 tablespoons sugar
1 tablespoon plus 2½ teaspoons ground cinnamon
1 tablespoon plus 1½ teaspoons instant espresso powder
1¾ cups all-purpose flour
2 teaspoons baking powder
Pinch of salt
¾ cup whole milk
2 large eggs
½ teaspoon pure vanilla extract
1 stick plus 2 tablespoons (10 tablespoons) unsalted butter, melted and cooled
3 ounces bittersweet chocolate, finely chopped, or ½ cup mini chocolate chips
For the Frosting:
6 ounces bittersweet chocolate, finely chopped
2½ tablespoons unsalted butter, cut into 4 pieces
1. Center a rack in the oven and preheat the oven to 350 degrees F. Butter an 8-inch square baking pan and line the bottom with parchment or wax paper. Place the pan on a baking sheet and set aside.
2. Stir 2 tablespoons of the sugar, 2½ teaspoons of the cinnamon and 1½ teaspoons of the espresso powder together in a small bowl.
3. In a large bowl, whisk together the flour, the remaining 1¼ cups sugar, the baking powder, salt and the remaining 1 tablespoon cinnamon. Mix the remaining 1 tablespoon of the espresso powder into the milk and warm the milk in a microwave until it is hot enough to dissolve the coffee. Cool the milk. In another bowl, whisk together the milk, eggs and vanilla. Pour the liquid ingredients over the flour mixture and gently whisk until you have a homogeneous batter. Now, using the whisk or a rubber spatula, fold in the butter with a light touch, just until the butter is absorbed. You'll have a smooth, satiny batter.
4. Scrape half of the batter into the pan and smooth the top. Sprinkle the chocolate over the batter and dust with the cinnamon-sugar mixture. Cover with the rest of the batter and smooth the top again.
5. Bake for 35 to 40 minutes, or until the cake is puffed and beginning to pull away from the sides of the pan; a thin knife inserted into the center will come out clean. Transfer the cake to a cooling rack and let it rest for 15 minutes before unmolding it onto another rack. Peel off the paper, invert it onto the first rack, and cool to room temperature right side up.
6. To make the frosting, put the chocolate and butter in a heatproof bowl and fit the bowl over a saucepan of simmering water. Cook, stirring gently and often, just until they melt. Be careful not to overheat the mixture so much that it things out; the chocolate should be smooth, very shiny, thick and spreadable. 9If it thins, leave the frosting at room temperature for a bit, until it thickens a little.)
7. Using an offset metal icing spatula or a table knife, spread the frosting in generous sweeps and swirls over the top of the cake. Allow the frosting to set at room temperature, then cut the cake into 9 squares, each about 2½ inches on a side. Wrapped in plastic, the cake will keep at room temperature for 2 days.