Zest lemons with a zester to get 3 tablespoons zest. Juice using a citrus juicer to get 2 tablespoons juice. Set aside the juice for the glaze.
Mix butter, sugar, and lemon zest in a stand mixer until creamy like peanut butter, about 2-3 minutes. Add eggs, vanilla, and lemon extract, and beat until combined. Scrape sides, then add baking powder, salt, and flour. Mix until no dry streaks remain.
Cover dough tightly and chill in fridge for 20 minutes.
Line a large plate with parchment paper. Add 1/2 cup powdered sugar to a small bowl.
Divide the dough into 1/4 cup-sized balls (about 60 grams; see note 5). Roll each ball in powdered sugar and place on parchment-lined plate. Repeat to use all the dough.
Chill dough balls for 20 minutes in fridge or 10 minutes in the freezer. Preheat oven to 325°F (162°C) and line a large sheet pan with parchment paper.
Bake 6 cookies at a time for 10–15 minutes until puffed and slightly golden around the edges. Cool on the pan for 5 minutes, then transfer to a wire rack to cool completely before glazing. Repeat with remaining dough balls.
Whisk 2 tablespoons lemon juice with 1 cup powdered sugar until smooth. Transfer to a zip-top bag, snip off the corner, and drizzle over cookies. Enjoy at room temperature or chilled.
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Notes
Note 1: I'll often use vanilla bean paste (3/4 teaspoon) instead of extract for more powerful flavor. Use vanilla extract or paste—whichever you prefer.Note 2: Lemon extract intensifies the citrusy flavor. Use pure lemon extract, not juice.Note 3: Correct flour measurement is very important. Too little flour leads to excessive cookie spreading; too much flour leads to dry and flavorless cookies. Use a food scale for accuracy; 400 grams is ideal.Note 4: The cookies work best at this size. While you may want smaller cookies, know that the texture will not be the same. The sweet spot for the dough balls is 60 grams of dough. Storage: Instead of freezing baked cookies, freeze the dough! Then transfer to a sealed container or bag for up to 3 months. Bake cookies straight from the freezer, adding a few extra minutes as needed. The glaze doesn't freeze and thaw well, so I recommend making it fresh.