Preheat the oven to 350 degrees F and line a baking sheet with a silicone mat. (You can also use parchment paper.) Beat the butter, sugar, vanilla and salt in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Add the egg whites and beat until incorporated. Reduce the mixer speed to low; add the flour and beat until just incorporated. Transfer 1 tablespoon of the batter to a small bowl and stir in the red food coloring.
Spoon 1 tablespoon of the plain batter onto the silcone mat; use an offset spatula to spread the batter into a very thin oval, about 4 by 6 inches. Repeat to make 3 more ovals on the mat. Dip a toothpick into the red batter and drag it diagonally across the ovals to make thin red stripes.
Bake until the cookies are set and the edges are just golden, about 7 minutes. Let cool 45 seconds, then immediately loosen the cookies from the baking sheet with an offset spatula and roll each one lengthwise around a chopstick or the handle of a thin wooden spoon to make a tube. Pull out the chopstick and transfer to a rack to cool completely. (You'll need to work quickly or the cookies will harden; if they do, return to the oven briefly to soften.)
Repeat with the remaining batter, letting the baking sheet cool completely between batches.