Melt butter for topping in a small saucepan over low heat. Stir in brown sugar until smooth. Spoon about 1 tablespoon of this mixture into the bottom of each muffin cup.
In a large bowl, beat the softened butter and granulated sugar until smooth. Add the egg and vanilla extract; mix until well combined.
Gradually add the dry ingredients to the wet mixture, alternating with the milk, and mix just until combined. Divide the batter evenly among the prepared muffin cups, filling each about 3/4 full.
Bake for 20-25 minutes, or until a toothpick inserted into the center comes out clean. Allow to cool in the tin for 5 minutes before inverting onto a cooling rack to release the mini cakes.