Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a large bowl, cream together the softened butter and granulated sugar until light and fluffy.
Beat in the eggs one at a time, then stir in the lemon zest, lemon juice, and vanilla extract.
In a separate bowl, whisk together the all-purpose flour, baking powder, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
Drop rounded tablespoons of dough onto the prepared baking sheets, about 2 inches apart.
Bake for 10-12 minutes, or until the edges are lightly golden brown. The centers should still be soft.
Let the cookies cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.
Once cooled, dust with powdered sugar if desired.