Blend the cottage cheese in a blender or food processor until completely smooth and creamy, about 1 minute. This gives the cookies the same soft, pillowy texture as classic Italian ricotta cookies.
In a large bowl, rub the lemon zest into the sugar with your fingers until fragrant. Add the softened butter and beat for 2 to 3 minutes, until light and fluffy.
Beat in the egg, then add the blended cottage cheese, lemon juice and vanilla and beat until smooth. The mixture may look slightly curdled, which is completely normal.
In a separate bowl, whisk the flour, baking powder, baking soda and salt, then mix into the wet ingredients on low speed just until combined. The dough will be very soft, so cover and chill it for 30 minutes.
Preheat the oven to 350°F and line 2 baking sheets with parchment paper. Scoop rounded 1 1/2 tablespoon portions of dough 2 inches apart.
Bake for 12 to 14 minutes, until the bottoms are lightly golden and the tops are set but still pale. Cool completely on a wire rack. Whisk the powdered sugar and lemon juice until thick but pourable. Dip the top of each cookie into the glaze, sprinkle with lemon zest and let set for 30 minutes before serving.