Preheat and Prepare: Preheat your oven to 350°F (177°C). Grease and line two 6-inch round cake pans with parchment paper.
Sift Dry Ingredients: In a bowl, sift together the cake flour, baking powder, and salt. Set aside.
Dissolve Espresso Powder: In a small measuring cup, dissolve the espresso powder in boiling water. Let it cool.
Cream Butter and Sugar: In a stand mixer fitted with the paddle attachment, cream the butter on medium speed until soft and creamy, about 2 minutes. Gradually add the sugar, continuing to beat until the mixture is pale and fluffy. Scrape down the bowl occasionally.
Add Vanilla and Eggs: Beat in the vanilla. Then add the eggs one at a time, fully incorporating each before adding the next. Scrape down the bowl as needed.
Alternate Dry and Wet Ingredients: On low speed, add the flour mixture in four parts, alternating with the milk. Start and end with the flour mixture, mixing until just smooth after each addition. Gently fold in the cooled espresso.
Bake: Divide the batter evenly between the prepared pans. Smooth the tops and bake in the center of the oven for about 20 minutes, or until a toothpick inserted comes out with a few moist crumbs. The edges should be pale golden, and the cake will start to pull away from the sides.
Cool: Allow the cakes to cool in the pans for 5 minutes, then turn out onto wire racks to cool completely, about 1 hour. Remove the parchment paper.