• Add the sugar and lemon zest to a large bowl. Rub the zest into the sugar with your fingertips for about 30 seconds, until the sugar feels slightly damp and smells very lemony.
• Add the softened butter and beat with a hand mixer for about 2 minutes, until pale and fluffy.
• Add the lemon juice, milk, and vanilla. Beat just until incorporated.
• In a separate bowl, whisk together the flour, cornstarch, baking soda, and salt.
• Add the dry ingredients to the butter mixture and mix on low or fold with a spatula just until a soft dough forms. Don't overmix.
• Cover and refrigerate the dough for 30 minutes.
• Preheat the oven to 350°F (180°C) and line baking sheets with parchment paper.
• Scoop about 1½ tablespoons of dough per cookie (a #40 scoop). Roll gently into balls and arrange about 2 inches apart.
• Press each dough ball down slightly into a thick disc. Don't flatten completely.
• Bake for about 11 minutes, until the edges are set but only barely beginning to color. The centers should still look soft.
• Cool on the baking sheet for 5–10 minutes, then transfer to a wire rack.

For the glaze (optional):

• Let the cookies cool completely.
• Whisk together the powdered sugar and 1 tablespoon lemon juice until smooth. Add milk, water, or more lemon juice 1 teaspoon at a time until thick but drizzleable.
• Drizzle or spread over the cooled cookies and top with a little fresh zest, if you like. Let the glaze set before storing.