Preheat your oven to 350°F (175°C) and line two baking sheets with parchment paper. Whisk together the flour, baking powder, and salt in a bowl.
Beat the softened butter and granulated sugar together until light and fluffy, about 3 minutes. Add the eggs one at a time, mixing after each. Stir in the fresh lemon juice, lemon zest, and vanilla extract.
Add the dry ingredients to the wet ingredients and mix on low speed until just combined. The dough will be soft and slightly sticky.
Scoop rounded tablespoon-sized portions of dough and roll them into smooth balls. Place them on the prepared baking sheets about 2 inches apart.
Bake for 10 to 12 minutes until the bottoms are just barely golden but the tops are still pale and soft. Do not overbake. These cookies should stay soft and cakey. Let them cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
Make the lemon glaze by whisking together the powdered sugar, fresh lemon juice, and lemon zest until smooth and thick but still pourable. Dip the top of each cooled cookie into the glaze, or drizzle it over with a spoon.Let the glaze set for about 20 minutes until firm. The bright lemony glaze on these soft, tender cookies is absolutely irresistible.