Preheat oven to 400 degrees. In a mixing bowl, whisk together flour, baking powder, salt, nutmeg and cinnamon, set aside.
In the bowl of an electric stand mixer fitted with the paddle attachment, whip together granulated sugar, brown sugar, butter and 1 Tbsp of the vegetable oil on medium-high speed until pale and fluffy.
Blend in remaining 3 Tbsp vegetable oil. Mix in eggs one at a time, blending just until combined after each addition and adding in vanilla and rum extract with second egg.
Add 1/3 of the flour mixture and using a rubber spatula fold into batter just until nearly combined, then add 1/2 of the eggnog and fold just until nearly combined, then repeat process once more and end by adding in last 1/3 of the flour mixture and folding just until combined (batter will be thick).
Divide batter among 12 paper lined muffin cups filling each cup full (about 1/3 cup in each).
Bake in preheated oven 18 - 21 minutes until toothpick inserted into center of muffin comes out clean. Allow to cool in muffin tin several minutes then transfer to a wire cooling rack.
Cool 10 minutes, then dip into glaze while still warm. Return to cooling rack and allow glaze to set at room temperature. Store in an airtight container.
For the glaze:
In a mixing bowl whisk together powdered sugar and 1/4 tsp nutmeg. Add in 3 Tbsp eggnog and rum extract and whisk to combined, mix in an additional tablespoon eggnog as needed (you want glaze to be fairly thick so it doesn't just run off).