Preheat your oven to 350°F (175°C). Line baking sheets with parchment paper.
In a large bowl, beat the softened butter and granulated sugar together until light and fluffy, which should take about 2–3 minutes.
Add the eggs one at a time, mixing well after each addition. Then stir in the vanilla extract, lemon extract, lemon juice, and lemon zest.
In another bowl, whisk together the flour, baking powder, and salt. Gradually add this dry mixture to the wet ingredients, mixing until just combined.
Use a tablespoon to scoop about 1 1/2 tablespoons of dough and roll them into balls. Place these balls on the prepared baking sheets, leaving about 2 inches between each.
Bake for 10–12 minutes, or until the edges are lightly golden. Let them cool on the baking sheets for 5 minutes before transferring them to a wire rack.
To make the glaze, whisk together the powdered sugar, lemon juice, and zest until smooth. Adjust the consistency with more lemon juice if needed.
Drizzle or dip the cooled cookies in the glaze and let it set before serving.