Heat oven to 325°F. Line a baking sheet with parchment paper or a silicone mat.
In a medium bowl, beat the softened butter, brown sugar, and granulated sugar until creamy and slightly fluffy, about 1 minute with a handheld mixer. Mix in the vanilla and espresso powder until smooth.
In a separate small bowl, whisk together the flour, black cocoa, baking powder, and salt. Add the dry mixture to the butter mixture and mix on low just until evenly combined; the dough will look sandy.
Sprinkle in 2 teaspoons milk and use your hands to gently press and fold the dough until it holds together without cracks. If still crumbly, add the remaining 1 teaspoon milk a few drops at a time.
Portion dough into 2-teaspoon mounds (about 12), roll into balls, and space them 2 inches apart on the prepared sheet. Using the flat bottom of a glass dipped in sugar, press each ball to roughly 1/4-inch thickness.
Bake 12 to 14 minutes, until edges are set and the tops look matte. Cool on the sheet for 5 minutes, then transfer to a rack to cool completely. Enjoy once crisp and set.