Preheat the oven to 400°F.
Line a standard size muffin tin with (12) paper liners.
In a mediumsized mixing bowl, cream together the butter, oil, and sugars until smooth.
Add the eggs, beating to combine.
Stir in the baking powder, baking soda, nutmeg, cinnamon, salt, and rum extract.
Stir the flour into the butter mixture alternately with the eggnog, beginning and ending with the flour and making sure everything is thoroughly combined.
Spoon the batter evenly into the prepared pan, filling the cups nearly full.
Bake the muffins for 15 to 17 minutes, or until they’re a pale golden brown and a toothpick inserted into the middle of one of the center muffins comes out clean.
In a medium bowl, prepare the glaze by mixing together the melted butter, confectioners’ sugar, vanilla and water. Whisk until smooth.
When muffins have cooled slightly, dip the muffin crown into the glaze and allow the glaze to harden.
Serve warm. Muffins will keep at room temperature for about a day.