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

Melt 1/2 cup butter, and allow to cool slightly. Add 3/4 cup room temperature champagne, 2 large eggs, 1 cup sugar and 1 teaspoon vanilla to the melted butter. Whisk to combine. Add 3/4 teaspoon salt and 2 teaspoon baking powder and give a quick stir, and then add 2 cups 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 1/4 cup champagne, 2 cups powdered sugar and 1/4 teaspoon 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.