Preheat oven to 350°F. Line two baking sheets with parchment paper.
Whisk flour, cocoa, baking soda, salt, and espresso powder in a bowl.
Cream softened butter with granulated and brown sugars until fluffy, 2–3 minutes.
Beat in vanilla and the beaten egg until smooth and glossy.
Stir in dry ingredients on low speed. Add milk and mix just until a soft dough forms.
Scoop level 1-tbsp portions, space 2 inches apart, and gently flatten to 2-inch rounds.
Bake 8–9 minutes, until edges set and centers look dry. Cool 5 minutes, then transfer to racks.
Beat filling butter until creamy. Mix in powdered sugar, vanilla, and salt. Add cream until thick and spreadable.
Pair similar-sized cookies. Spread or pipe 1 tbsp filling on a cookie, top with another, and press gently.