Preheat the oven to 350 degrees F, and grease two doughnut pans. Set aside.

Melt the butter in a large microwave safe mixing bowl, and allow to cool slightly. Add the champagne, eggs, sugar and vanilla to the melted butter. Whisk to combine. Add the salt and baking powder and give a quick stir, and then add flour. Stir with a spatula until no flour streaks show in the batter.

Transfer the batter to a large gallon-size ziplock bag, and cut a small hole in the corner. Pipe the batter into each well of the greased doughnut pan, filling to about 1/4 inch from the top of the mold. Bake 14-16 minutes, or until a toothpick comes out with moist crumbs. Allow to cool in the pans for 5 minutes, and then transfer to a cooling rack to cool completely. In the meantime, prepare the glaze.

For the glaze: Combine the champagne, powdered sugar and salt in a shallow bowl, whisking to combine. Generously dip each cooled doughnut into the glaze, and then return to the cooling rack for the glaze to set.