In a small mixing bowl, whisk together the egg whites, sugar and vanilla. Stir in the salt and half of the flour, then mix in the melted butter. Beat in the rest of the flour until smooth.
Line a baking sheet with parchment paper. Use the back of a big spoon to spread 2 level tablespoons of the batter into a circle 6 inches across. Try to get the circles as even and smooth as possible. You're likely to get 2 rounds on one standard baking sheet.
Put the baking sheet in the oven, and begin checking the cones after about 10 minutes. Depending on your oven, they'll take between 10 and 15 minutes to bake. The circles should be golden brown throughout (some lighter and darker spots are inevitable so don't worry).
Remove the baking sheet from the oven. Use a thin metal spatula to loosen the edge of one disk. Slide the spatula under the disk, quickly flip it over, and immediately roll it around a cone form Press the seam firmly on the counter to close the cone and pinch the point at the bottom securely closed.
Let the cone cool slightly on the mold until it feels firm, then slide it off and stand it upright in a tall glass to cool. Roll the other cones the same way. Repeat, using the remaining batter. Place in cone holder of some sort to cool. Glasses work well.
Serve up with one or two scoops of your favorite ice cream.