Preheat oven to 350 degrees and grease springform pan.
Cream 6 tablespoons of butter with the granulated sugar until light and fluffy for two to three minutes. On low speed, add the eggs one at a time, beating well between each addition.
6 tablespoons unsalted butter, 3/4 cup sugar, 2 eggs
Add 2 tablespoons of espresso, the vanilla, and the yogurt and mix until thoroughly combined. Add the baking soda, baking powder, 1/2 teaspoon of salt, and 2 teaspoons of cocoa powder and mix again until thoroughly combined. Add 1 1/2 cups of the flour and mix until just incorporated. Pour batter into your prepared cake pan.
In a new bowl, place the brown sugar, cinnamon, nutmeg, 2 teaspoons of cocoa powder, 1/2 teaspoon salt, and the last 1/4 cup of flour. Mix well, then add 2 tablespoons of the espresso and 2 teaspoons of melted butter and stir.
1/2 cup brown sugar, 2 teaspoons cinnamon, 1/2 teaspoon nutmeg, 2 teaspoons cocoa powder, 1/2 teaspoon salt, 1/4 cup flour, 2 tablespoons brewed espresso, 2 teaspoons unsalted butter
Drop dollops of the mocha mixture onto the cake. Dip a fork into the cake and swirl to distribute the mocha mixture.
Bake for 45-50 minutes, until a cake tester comes out clean. Cool completely.
To make the drizzle, combine 2 tablespoons melted butter, milk, and the last 3 tablespoons of espresso. Add powdered sugar, and mix until it's smooth.