In the bowl of a stand mixer with the paddle attachment, beat the butter, brown sugar, and 1/4 cup of the white granulated sugar on medium speed until light and fluffy.
With the paddle running, add the vanilla, then the eggs one at a time.
In a separate bowl, whisk together the flour, baking powder, baking soda, and salt. Alternate pouring the dry ingredients into the muffin mixture with the buttermilk in turn, beginning and ending with the dry mixture.
Line a regular-sized muffin tin with muffin liners, or spray so the muffins won't stick in the tin. Fill each muffin mold about 3/4 of the way full. Bake in an oven preheated at 350 degrees Fahrenheit for 23 minutes, or until they are set and golden around the edges. Let them cool until they are easy to handle.
To top the muffins, melt the stick of butter in the microwave or over the stove top, and place in a shallow bowl. Stir the 1/2 cup of sugar and 2 teaspoons of cinnamon until well combined in a separate shallow bowl. Dip the entire muffin top into the melted butter, then roll in the cinnamon sugar mixture until well coated.