Preheat oven to 375°F. Line baking sheets with parchment paper.
Process cottage cheese in food processor or blender until mostly smooth. Reserve.
Combine flour and baking powder. Set aside.
Using an electric mixer beat butter and sugar until light and fluffy (about 3 minutes). Add egg and beat until incorporated. Add cottage cheese, ½ lemon zest, ½ lemon juice, and vanilla. Beat to combine.
Stir in dry ingredients until just combined.
Spoon 2 tablespoons dough for each cookie onto lined baking sheet. Bake 15 minutes, until slightly golden. Remove from oven and let rest on baking sheet for 20 minutes.
Meanwhile, combine powdered sugar, remaining lemon zest, and remaining lemon juice. Stir until smooth. Spoon about 1/2 teaspoon onto each cookie and use spoon to spread gently. Let glaze harden (about 2 hours) before serving.