Position oven rack in center of oven; preheat oven to 350°.
Butter two 9 x 2 inch round cake pans; dust the insides with flour, tap out the excess; line the bottoms of the pans with parchment paper or wax paper; put the pans on a baking sheet.
Make the cake: sift together the cake flour, baking powder, baking soda, and salt.
Using a stand mixer fitted with paddle attachment, beat the butter on medium speed until soft and creamy.
Add in sugar and beat for 3 minutes.
Add the eggs one at a time and then the yolk, beating for 1 minute after each addition.
Beat in the vanilla (don’t be concerned if the mixture looks curdled).
Decrease mixer speed to low and add the dry ingredients alternately with the buttermilk, adding the dry ingredients in 3 additions and the milk in 2 (begin and end with dry ingredients); scrape down sides of bowl as needed and mix only until the ingredients disappear into the batter.
Divide the batter evenly between the two pans and smooth the tops with a rubber spatula.
Bake for 28-30 minutes, rotating the pans at midway point.
When fully baked, the cakes will be golden and springy to the touch and a thin knife inserted into the centers will come out clean.
Transfer cakes to a rack and cool for about 5 minutes, then run a knife around the sides of the cakes, unmold them and peel off the paper liners.
Invert and cool to room temperature right side up.
Make the extract: In a small bowl or cup, stir the espresso powder and boiling water together until blended; set aside.
Make the syrup: In a small saucepan, stir the water and sugar together; bring just to a boil.
Pour the syrup into a small heatproof bowl and stir in 1 tablespoon of the espresso extract and the amaretto; set aside.
Make the filling/frosting: add mascarpone, sugar, vanilla, and liqueur in a bowl; whisk just until blended and smooth.
Using a stand mixer with whisk attachment, whip the heavy cream until firm peaks form.
Switch to a rubber spatula and stir about ¼ of the whipped cream into the mascarpone.
Fold in the rest of the whipped cream with a light touch.
Assemble the cake: if the tops of the cake layers have crowned, use a long serrated knife and a gentle swaying motion to even them.
Place one layer right side up on a cardboard round or a cake plate protected with strips of wax or parchment paper.
Using a pastry brush, soak the layer with about 1/3 of the espresso syrup.
Smooth some of the mascarpone cream over the layer—use about 1 ¼ cups—and gently press the chopped chocolate into the filling.
Put the second cake layer on the counter and soak the top of it with half the remaining espresso syrup, then turn the layer over and position it, soaked side down, over the filling.
Soak the top of the cake with the remaining syrup.
For the frosting—whisk 1 to 1 ½ tablespoons of the remaining espresso extract into the remaining mascarpone filling.
Taste the frosting as you go to decide how much extract you want to add.
If the frosting looks as if it might be a little too soft to spread over the cake, press a piece of plastic wrap against its surface and refrigerate it for 15 minutes or so; refrigerate the cake too.
With a long metal icing spatula, smooth the frosting around the sides of the cake and over the top.
Decorate the cake with chocolate-covered espresso beans—press them into the filling, making concentric circles of beans or just putting some beans in the center of the cake.
Refrigerate the cake for at least 3 hours before serving—the elements need time to meld.
Just before serving—dust the top of the cake with cocoa.
To serve—let sit at room temperature for 20 minutes, cut with a serrated knife and a gentle sawing motion; serve with coffee or espresso.