First, reduce 1 cup of champagne to 1/2 cup by boiling in a small pan on stovetop. Let cool.
Preheat oven to 350 degrees F. Line muffin tin with paper liners.
Cream the butter and sugar at medium speed, then reduce to low to add the eggs one at a time.
Add the reduced champagne.
Sift together the flour, baking powder, and salt.
Alternate adding the flour mixture and the yogurt, beginning and ending with the dry ingredients. Mix until just combined, do not overmix.
Scoop into lined tins and bake for about 20 minutes (10-12 minutes for mini cupcakes), turning halfway through.
Remove from oven and let cool in pans for 5 minutes. Cool on racks until room temperature before frosting.
For frosting: Beat the butter until smooth. Slowly add half the powdered sugar, then 3 tablespoons reduced champagne. Add the rest of the powdered sugar and mix, adding more champagne if needed to reach desired consistency.