1. In a bowl, with a mixer on high speed, beat butter, powdered sugar, peppermint extract, and vanilla until well blended. Beat in egg whites until incorporated. Reduce speed and mix in flour, salt, and chocolate chips. Cover bowl with plastic wrap and chill until dough is firm but pliable, about 30 minutes.
2. With powdered sugar-dusted hands, shape dough into 1 1/2-inch balls, flattening each slightly between your palms to about 2 1/2 inches wide. Place 1 inch apart on buttered 12- by 15-inch baking sheets.
3. Bake in a 300° regular or convection oven until cookies feel firm to touch and are barely beginning to turn golden on the edges, 10 to 12 minutes. If baking more than one sheet at a time in one oven, switch positions halfway through baking. Transfer cookies to racks to cool completely.
Nutritional analysis per cookie.