Preheat oven to 350° F. Line baking sheets with parchment paper.
In a medium mixing bowl, whisk together the flour, baking powder, baking soda and salt.
2 ¾ cups (330g) all-purpose flour*, ½ teaspoon baking powder, ½ teaspoon baking soda, ½ teaspoon salt
Add the granulated sugar and lemon zest to the bowl of a stand mixer. Use your fingertips to work the zest into the sugar until fragrant and well distributed.
1 ¼ cups (250g) granulated sugar, 2 tablespoons fresh lemon zest
Add the butter to the sugar-zest mixture and mix until creamy, starting on low speed and then increasing the speed to medium-high. Add the egg, egg yolk, lemon juice and vanilla and mix in on medium speed until incorporated, scraping down the sides of the bowl as needed.
¾ cup (170g) unsalted butter, 1 large egg, 1 egg yolk, 2 tablespoons fresh lemon juice, 1 teaspoon vanilla extract
Add the dry ingredients to the wet and mix on low speed until combined. It will be a fairly firm dough.
Roll the dough into 1-inch balls.* Place on parchment-lined baking sheets, about 3 inches apart. Bake one pan of cookies at a time, for 8-10 minutes, until tops are almost set and bottoms are light golden brown. For soft and chewy cookies, be careful not to overbake. (The cookies may still be slightly puffy when you take them out of the oven and will flatten more as they cool on the baking sheet.)
Let the cookies cool fully before adding the glaze.
Once cookies are completely cool, make the glaze: In a small bowl whisk together the powdered sugar, lemon zest and lemon juice. Dip the top of each cookie in the glaze, hold upside down for a few seconds to let the excess glaze drip off, then set cookies on a sheet of parchment paper to let the glaze set (it will take a few hours to set and will still be slightly soft). You can speed up the process by putting the glazed cookies on parchment lined baking sheets and chilling in the refrigerator for about an hour.
1 ½ cups (180g) powdered sugar, 3 tablespoons fresh lemon juice, 1 teaspoon fresh lemon zest