Cream sugar, butter, and lemon zest together in a large bowl using an electric mixer until light and fluffy, about 3 minutes.

Combine eggs, lemon juice, and vanilla extract in a cup; mix using a fork until combined. Beat into egg mixture slowly on low speed.

Whisk flour, baking powder, and salt together in a medium bowl. Add to egg mixture slowly on low speed until just combined. Do not overbeat; dough will be sticky. Cover and chill dough in the refrigerator for 30 minutes; it will still be sticky after chilling.

Preheat the oven to 350 degrees F (175 degrees C). Line 2 large baking sheets with parchment paper.

Form dough into balls using a cookie scoop or tablespoon and roll each in powdered sugar. Place onto the prepared baking sheets, spacing 2 inches apart. Lightly smash each ball down using your hand.

Bake in the preheated oven until the middle of each cookie is just barely set, 11 to 12 minutes. Let cool on a wire rack completely before storing in an airtight container.