Warm the champagne and dissolve the yeast in it.

Add milk, sugar, salt, eggs, butter, champagne flavor, and 2 cups of flour.

Beat on low for 30 seconds with the paddle attachment, scraping the bowl if needed.

Beat on medium speed for 2 minutes, scraping bowl occasionally.

Switch to the dough hook attachment and knead in the remaining 3 cups of flour until smooth.

Cover the bowl with a dishcloth and let the dough rise until it has doubled in size, about an hour.

Turn the dough onto a floured surface and roll it 1/2-inch thick.

Cut with a floured doughnut cutter or number cutters.

Cover and let rise again until doubled in size, 30-40 minutes.

Place a layer of paper towels next to the stove.

Heat vegetable oil in a deep fryer to 350 degrees F and fry until golden brown, about 30 seconds on each side.

Remove from the oil and drain on paper towels for a minute.

Dip the doughnuts into the glaze and set them on cooling racks over parchment or foil-lined baking sheets.

If you run out of glaze, scrape the excess off the baking sheet and return it to the bowl.

Heat the butter until melted over medium-low heat and remove from the stove.

Whisk in the powdered sugar, champagne, and champagne flavor until smooth.

If the glaze is too thick to dip the doughnuts into, thin it out by adding more champagne one teaspoon at a time.

Dip the doughnuts into the glaze. Transfer them to cooling racks over parchment or foil-lined baking sheets.

Sprinkle with the edible decorations.

If you run out of glaze, scrape the excess off the baking sheet and return it to the bowl.

Let the glaze harden completely before serving.