Process almonds in 2 batches in a food processor or blender until finely ground.
Combine ground almonds, powered sugar, and vanilla extract in a medium bowl, using hands to mix. Stir in egg whites. (Dough will be sticky.) Drop by teaspoonfuls onto lightly greased baking sheets.
Bake at 350° for 15 to 20 minutes or until lightly browned.