Place a rack in the upper third of the oven and preheat oven to 350 degrees F. Lightly grease a mini ddoughnut pan and set aside.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, nutmeg, and sugar. Set aside.
In a small saucepan over medium-low heat, melt butter. Butter will begin to crackle and pop as it melts. That’s the water melting out of the butter. Once the water has evaporated the butter will quiet down and begin to brown. Keep an eye on it, it browns quickly. The butter will begin to smell nutty. Remove from heat and immediately transfer browned butter (brown bits and all) to a small bowl.
In a small bowl whisk together egg,milk, and vanilla extract. Whisk in the browned butter, brown bits and all. Whisk in the champagne.
Add the wet ingredients all at once to the dry ingredients. Stir together until no flour bits remain and all of the ingredients are well combined. Try not to overmix the batter. That might create rubbery doughnuts.
Use a small spoon to dollop batter into the prepared pan. Smooth out and fill each doughnut in the pan three-quarters full with batter.
Place in the oven and bake for 8 to 10 minutes. Keep an eye on them and try not to over-bake them. Remove from the oven and allow to cool in the pan before inverting onto a wire rack to cool completely.
While the doughnuts cool, make the glaze.
To make the glaze, in a medium bowl whisk together powdered sugar and salt. I didn’t sift my powdered sugar. I was able to whisk the lumps out. If you have stubborn powdered sugar, sifting before is best.
Add melted butter, milk, and vanilla extract. Whisk to combine. Add champagne as necessary to create a thick but still just pourable glaze.
Once the doughnuts are completely cool, dip top-side-down into the glaze. Return to the wire rack and sprinkle with toppings. Allow to set for about 30 minutes before stacking or serving. Doughnuts are best within 2 days.