Preheat the oven and grease three 8-inch cake pans, lining the bottoms with parchment paper for easy removal.
Whisk the flour, sugar, dark cocoa powder, baking powder, baking soda, and salt in a large bowl until no lumps remain.
Stir the instant espresso into warm water, then mix in the sour cream, oil, eggs, and vanilla in a separate container.
Pour the wet mixture into the dry ingredients and whisk gently until just combined, then divide the batter into the pans.
Bake until a toothpick comes out clean, then let the cakes cool while preparing the syrup and frosting.
Simmer the water, sugar, and cocoa powder for the syrup in a small pan until the sugar dissolves, then set aside to cool.
Beat the butter and cocoa together, then gradually add powdered sugar, cream, and melted chocolate until the frosting is light and fluffy.
Level the cooled cakes, brush each layer with syrup, and stack them with thick layers of buttercream in between.
Frost the outside of the cake and press mini chocolate chips into the sides before piping decorative swirls on top.