Whisk the all-purpose flour, baking powder, and fine salt together in a medium bowl.
Beat the softened unsalted butter and granulated sugar until pale and creamy.
Beat in the egg and vanilla extract until the mixture is smooth.
Gradually mix the dry ingredients into the butter mixture just until a soft dough forms.
Remove one quarter of the dough, knead the unsweetened cocoa powder into it, and chill both doughs for 15 minutes.
Heat the oven to 350°F, shape the vanilla dough into 24 rounds, and press small cocoa dough pieces onto each round to form one large pad and four toes.
Bake for 10 to 12 minutes, then cool the cookies on the pan for 5 minutes before transferring them to a rack.
Notes
Keep the cocoa dough pieces small so the paw-print details remain distinct as the cookies spread.