Preheat the oven to 250 degrees. Line two sheet pans with parchment paper or silicone baking mats.
In a large mixing bowl, beat the egg whites, cream of tartar, and the salt until stiff. Continue beating, adding the sugar a little at a time, until you have glossy peaks. Beat in the vanilla extract and the lemon zest.
Dollop the mixture onto the prepared baking sheets, leaving at least an inch between cookies. Bake for 45 minutes. Allow the cookies to dry in the oven for several hours or overnight. Once cooled, store in an airtight container.
Store leftover meringue cookies in an airtight container at room temperature for 3 weeks.