Preheat oven to 400 degrees. In a mixing bowl, add butter, oil, sugar, brown sugar, vanilla, and rum extract. Cream together. Blend in eggs.
In another mixing bowl, add flour, baking powder, baking soda, salt, and nutmeg.
Measure eggnog. Add to creamed mixture alternately with dry ingredients. Mix just until combined.
Using standard muffin cups, line muffin pan and fill 2/3 full. If you don't have muffin cups, you can butter and flour the muffin pan.
Bake on middle rack of oven at 400 degrees for 15 to 18 minutes. Baking time will vary depending o your oven. Make the glaze while muffins bake. Test muffins for doneness with a toothpick. Remove from pan immediately to wire rack to cool.