12 small solid chocolate Easter eggs, unwrapped and frozen
1 1/2 cups all-purpose flour
1 3/4 teaspoons baking powder
1 cup white sugar
1/2 cup butter, softened
2 eggs
2 teaspoons vanilla extract
1/2 cup milk
 
1/2 cup butter, softened
2 cups confectioners' sugar
1/2 teaspoon vanilla extract
1 tablespoon milk, or as needed
1 drop food coloring, any color, or as needed
12 candy-coated chocolate eggs (such as Cadbury Mini Eggs®)