Prepare the cake pans
Preheat the oven to 350°F. Grease two 8-inch round cake pans with nonstick baking spray. For easier removal, line the bottoms with parchment paper.
Beat the eggs and sugar
Add the eggs and sugar to a large mixing bowl. Beat for about 3 minutes, until the mixture looks lighter, frothy, and slightly thickened.
Add the wet ingredients
Mix in the Greek yogurt, vegetable oil, vanilla extract, and espresso powder until smooth and combined.
Whisk the dry ingredients
In a separate bowl, whisk together the gluten-free flour, cocoa powder, baking powder, baking soda, and salt.
Combine wet and dry
Add the dry mixture to the wet mixture in batches. Mix gently until just combined. Do not overmix, or the cake can turn dense.
Stir in the coffee
Slowly pour in the brewed coffee and stir until the batter is smooth. The batter may look thinner than traditional cake batter, and that is okay.
Bake the cake layers
Divide the batter evenly between the prepared pans. Bake for 35 to 45 minutes, or until a toothpick inserted in the center comes out clean or with a few moist crumbs.
Cool completely
Let the cakes cool in the pans for about 10 minutes, then transfer them to a wire rack. Cool fully before frosting.
Make the chocolate buttercream
Add the softened butter, powdered sugar, cocoa powder, espresso powder, milk, and vanilla to a mixing bowl. Beat until smooth, fluffy, and creamy. Add a little more milk if the frosting feels too thick.
Frost the cake
Place one cooled cake layer on a serving plate. Spread a thick layer of chocolate buttercream over the top. Add the second cake layer, then frost the top and sides of the cake.
Slice and serve
Let the cake sit for a few minutes so the frosting settles, then slice and enjoy.