Cream butter, granulated sugar, and brown sugar until fluffy. Beat in eggs and vanilla. In another bowl, whisk flour, baking powder, and salt, then gradually add to the wet mixture. Mix until combined. Cover and chill for 30 minutes.
Preheat oven to 350°F (175°C) and line a baking sheet with parchment paper. Mix brown sugar, cinnamon, and melted butter. Scoop 3 tablespoons of dough, form into a ball, press a thumbprint, and add ½ teaspoon of filling. Seal dough and flatten into thick discs.
Place cookies on the sheet, spacing them apart. Bake for 12–14 minutes until edges are golden and centers are soft. Cool on a wire rack.
Beat softened cream cheese and butter until smooth. Add powdered sugar and vanilla, mixing until fluffy. Gradually add cream or milk, one teaspoon at a time, until smooth.
Pipe frosting over cooled cookies in a swirl pattern. Sprinkle with cinnamon or brown sugar for an extra touch.