Add 3/4 cup packed light or dark brown sugar and 1/4 teaspoon kosher salt to the bowl with the butter. Beat with the paddle attachment (or with electrix mixer beaters) on high speed until light and fluffy, 2 to 3 minutes. Scrape down the sides of the bowl. Add the egg, 1/2 cup applesauce, and 2 teaspoons vanilla extract, and beat on low speed until combined (the mixture will look curdled).