Preheat the oven to 350 degrees F. Line the bottoms of three 8-inch round cake pans with parchment paper and grease the sides with nonstick cooking spray.

In a stand mixer fitted with the whisk attachment, beat the butter and vegetable oil on low speed until combined, then increase speed to medium-high and beat until smooth, about 2 minutes.

With the mixer running, slowly pour in the granulated sugar and beat on medium-high until the mixture is light, fluffy, and pale yellow, about 2 minutes.

Scrape down the sides and bottom of the bowl, then add the eggs and vanilla extract. Beat on medium-high until light and fluffy, about 2 minutes, scraping down the bowl as needed.

In a separate bowl, whisk together the flour, cocoa powder, espresso powder, baking powder, and salt. In a third bowl, combine the milk and coffee liqueur. With the mixer on low speed, alternate adding the dry ingredients and the wet ingredients in 3 additions.

Divide the cake batter evenly among the prepared pans. Bake 20 to 25 minutes, until tops spring back when touched and testers inserted in the centers come out clean. Cool slightly on racks, then remove the cakes from the pans by dragging a butter knife around their edges, inverting onto plates, then flipping onto racks to cool completely, about 30 minutes.

In a small saucepan over medium heat, combine the granulated sugar, coffee liqueur, and vodka. Bring to a simmer, stirring constantly, until the mixture thickens and coats the back of a silicone spatula, about 5 minutes. Remove from the heat and stir in the vanilla extract. Set aside to cool, about 30 minutes.

In a stand mixer fitted with the paddle attachment, combine the butter and confectioners’ sugar and beat on low for 30 seconds. Add the espresso powder, milk, coffee liqueur, vanilla extract, and salt. Beat on medium, scraping down the sides and bottom of the bowl as needed, until the frosting is fluffy, about 1 minute.

Place the first cake layer on a cake stand or plate. Use a pastry brush to soak the top of the cake with about one-third of the coffee syrup, gently pressing the brush into the cake to help it absorb. Top with a large dollop of frosting, spreading it evenly to the edges of the cake.

Repeat with the second and third cake layers. Cover the sides of the cake with the remaining frosting. If desired, top with espresso beans.