Preheat an oven to 350°F Grease 9 standard muffin cups with butter or butter-flavored nonstick cooking spray; fill the unused cups one-third full with water to prevent warping.
To make the muffins, in the bowl of an electric mixer fitted with the flat beater, combine the butter and sugar and beat on medium speed until light and fluffy. Add the egg and beat well until pale and smooth.
In another bowl, stir together the flour, baking powder, baking soda, salt and nutmeg. Add to the butter mixture in 2 additions, alternating with the buttermilk and vanilla. Stir just until evenly moistened. The batter will be slightly lumpy.
Spoon the batter into the prepared muffin cups, filling each three-fourths full. Bake until the muffins are golden, dry and springy to the touch and a toothpick inserted into the center of a muffin comes out clean, 20 to 25 minutes. Transfer the pan to a wire rack and let cool for 5 minutes. Unmold the muffins and let stand until cool enough to handle.
To make the topping, in a small, shallow bowl, stir together the sugar and cinnamon. Put the melted butter in another small bowl. Holding the bottom of a muffin, dip the top into the melted butter, turning to coat it evenly. Immediately dip the top in the cinnamon-sugar mixture, coating it evenly, then tap it to remove excess sugar. Transfer the muffin, right side up, to the rack. Repeat with the remaining muffins. Let cool completely before serving. Makes 9 muffins.
Note: These cakelike muffins are made from a batter similar to that used for preparing doughnuts, but they are baked instead of fried.