Preheat the oven to 325°F (165°C). Line 3 baking sheets with parchment paper. In a large bowl, combine the egg whites and cream of tartar. Using a mixer on medium-high speed, beat until the egg whites are very foamy, about 1 minute. Add a pinch of salt. While beating continuously, gradually add the sugar and beat until stiff peaks form, 3–4 minutes.
Using a rubber spatula, stir in the vanilla. In 3 batches, gently fold the coconut into the beaten whites just until incorporated. Place rounded tablespoonfuls of the dough 1½ inches (4 cm) apart on the prepared baking sheets. Bake until the cookie edges begin to turn light golden brown, 19–22 minutes.
Let the cookies cool completely on the sheets, about 30 minutes. Store in a single layer in an airtight container at room temperature for up to 4 days.