Preheat oven to 350 degrees F. Line a large baking sheet with parchment paper or a silicone baking mat and set aside.
In a bowl whisk together the flour, sugar, and salt. Whisk in the egg whites, butter, and vanilla until well combined. Let the batter sit at room temperature for 45 minutes. If using later, cover the surface with plastic wrap and refrigerate.
Drop rounded tablespoons of the batter about 6 inches apart onto the baking sheet. Spread the batter with an offset spatula or the back of a spoon into 5 or 6-inch rounds.
Bake in the middle of the oven until golden and lightly brown around the edges, about 6 to 8 minutes. To shape, gently loosen a cookie from the baking sheet with an offset spatula, then turn it over so the smooth side is facing up. Roll the cookie into a cone and hold the bottom of the cone together for several seconds until cookie holds its shape. Carefully slip the cone into a Champagne flute and let it cool completely. Shape the remaining cookies. If the cookies become too brittle to roll, return the baking sheet to the oven for a few seconds to soften. Make and shape more cookies with the remaining batter.
Chef's Note: If the bottoms of the cones do not close tightly, fill them with peanut butter, chopped candy, or melted chocolate that has been cooled.