Preheat oven to 350 F. Butter an 8-inch square baking pan and line the bottom with parchment or wax paper. Place the pan on a baking sheet.

To Make the Cake, stir 2 tablespoons of the sugar, 2 1/2 teaspoons of the cinnamon and the espresso together in a small bowl; set aside.

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

In a small bowl, whisk together the milk, eggs and vanilla. Pour the liquid ingredients over the flour mixture and gently whisk until you have a homogenous batter. Using a rubber spatula, fold in the butter with a light touch, just until the butter is absorbed. You will have a smooth, shiny batter.

Scrape half of the batter into the pan and smooth the top. Sprinkle the chocolate and cinnamon chips over the batter and dust with the cinnamon-sugar mixture. Cover with the rest of the batter and smooth the top again.

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.

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 thins out; the chocolate should be smooth, very shiny, thick and spreadable. (If it thins, leave the frosting at room temperature for a bit, until it thickens a little.)

Using an offset metal icing spatula, spread the frosting in generous sweeps and swirls over the top of the cake. Allow the frosting to set at room temperature before serving.