Heat the oven to 350 F with a rack in the middle position. Line a 12-cup muffin tin with muffin liners.

Whisk together the flour, baking powder, baking soda, salt, nutmeg and allspice. Set aside.

In the bowl of a stand mixer fit with a paddle attachment (or with a hand mixer), beat the butter on medium-high speed until creamy, scraping down the sides as needed. Add the sugar and beat until light, fluffy, and creamy, again scraping down the sides as needed. Add the eggs one at a time, beating until the batter is smooth.

Remove the mixing bowl from the mixer and use a stiff spatula for these next steps. Add a quarter of the flour mixture to the bowl and mix until just barely combined. Mix in 1/3 of the eggnog plus vanilla until the batter smooths out again. Continue adding the flour mixture and the eggnog alternately, ending with the last quarter of the flour mixture. At this point, the batter will be fairly thick. Try not to overmix, but make sure all the ingredients are fully incorporated.

Divide the batter between the muffin tins, filling each one almost to the top. Bake until the muffins have puffed up, are starting to brown around the edges, and a cake tester comes out clean, 25 to 30 minutes. Transfer the muffins to a wire cooling rack.

When the muffins are cool enough to handle but still warm, melt the butter in a microwave-safe dish (or on the stovetop) and use a pastry brush to paint the tops of the muffins with butter.

Sift powdered sugar over the doughnut muffins. For a heavy coating, do this twice.