Mix Dough: Beat the butter with an electric mixer for 2 minutes until fluffy. Add the sweetened condensed milk and sugar; mix until combined. Mix in the egg and vanilla, scraping the bowl occasionally.
Combine Dry Ingredients: Add the flour, cocoa powder, baking powder, and salt. Mix on low speed until mostly combined. Finish mixing by hand with a spatula to ensure everything is incorporated.
Chill Dough: Wrap the dough tightly in plastic wrap and refrigerate for at least 1 hour, or overnight.
Preheat Oven: Set the oven to 350°F.
Roll and Cut: Divide the chilled dough into 4 pieces. Roll out one piece at a time to ¼-inch thick on a lightly floured surface. Use cookie cutters to cut shapes and place cookies 1 inch apart on a parchment-lined baking sheet.
Bake: Bake for about 8 minutes, depending on cookie size. When the centers look set and not shiny, they’re done. Repeat with remaining dough. Store baked cookies in an airtight container at room temperature for up to 4 days or freeze for 1-2 months.