Preheat oven to 350°F and line two baking sheets with parchment. Beat butter and sugar until fluffy, about 2 minutes. Add eggs one at a time, then vanilla and 1 tsp lemon zest. In a separate bowl, whisk flour, baking powder, and salt. Add to wet mixture in two batches, mixing just until combined.

Roll rounded tablespoons of dough into 5-inch ropes. Tie each into a knot, tuck ends under, and place on baking sheets, spacing cookies an inch apart.

Bake one sheet at a time for 12 to 14 minutes, until bottoms are lightly golden. Cool on the sheet 5 minutes, then transfer to a rack to cool completely.

Whisk powdered sugar, 2 tbsp lemon juice, and 1 tsp lemon zest until smooth. Add a few drops more juice if needed for a thick but pourable consistency.

Dip the top of each cooled cookie into the glaze, let excess drip off, and place on a rack. Let sit until glaze is firm, about 15 minutes.