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: Sift the cake flour, baking powder, and salt together in a bowl. Set aside.
Dissolve Espresso Powder: In a small measuring cup, dissolve the espresso powder in boiling water and let it cool completely.
Cream Butter and Sugar: Using 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, beating until the mixture is pale and fluffy. Scrape down the sides of the bowl occasionally.
Add Vanilla and Eggs: Beat in the vanilla extract. Add the eggs one at a time, ensuring each is fully incorporated before adding the next. Scrape down the bowl as needed.
Alternate Dry and Wet Ingredients: Reduce the mixer speed to low and add the dry flour mixture in four parts, alternating with the milk. Begin and end with the dry ingredients, mixing until just combined after each addition. Finally, gently fold in the cooled espresso.
Bake: Divide the batter evenly between the prepared pans and smooth the tops. Bake in the center of the oven for 20 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs. The edges should be pale golden, and the cake will slightly pull away from the sides of the pans.
Cool: Let the cakes cool in the pans for 5 minutes, then turn them out onto wire racks. Peel off the parchment paper and allow them to cool completely, about 1 hour.