Preheat the oven to 350°F. Line two muffin tins with paper liners and set aside.
Cookie dough
In a medium bowl, cream together the ½ butter and brown sugar until light and fluffy. Add in the egg and 1 teaspoon vanilla, and mix until fully combined. Mix in the 1 ⅓ cup flour, baking soda and salt until fully incorporated.
Using a medium cookie scoop (around 1.5 Tablespoons), scoop the cookie dough into the bottom of the muffin cups, and press down with the back of a spoon.
Cupcakes
In a medium bowl, cream together the ½ cup butter and white sugar until light and fluffy. Add in the 2 eggs and 1 Tbs vanilla until fully combined.
In a separate bowl, mix together the 1 ½ cup flour and baking powder.
Alternate between mixing in the flour mixture and the milk into the wet ingredients, starting and ending with the flour.
Divide the batter evenly on top of the cookie dough, and bake in the preheated oven for 18-23 minutes, or until a toothpick inserted comes out clean.
Frosting
In a medium bowl, cream the butter until smooth and creamy. Slowly add in the powdered sugar, and mix until combined. Add in the 1 Tbs vanilla, and mix until fully incorporated. If the frosting is too thick, add in the milk.