Place butter, ⅔ cup granulated sugar and brown sugar, and lemon zest in a large bowl. Beat with a hand or a stand mixture for 2-3 minutes, or until creamed and fluffy.
Mix in egg, lemon extract, baking soda and salt, then mix in flour until cookie dough forms.
Place remaining ¼ cup granulated sugar in a small bowl. Scoop 1 tablespoon sized cookie dough balls and roll in sugar, then place on a cookie sheet covered with parchment paper or a silicone baking mat. No need to space; we are chilling the dough.
Chill the dough for at least 30 minutes.
Preheat oven to 350°F. Line another cookie sheet then spread cookies at least 2-inches apart. Bake for 11-14 minutes or until they just lose their glossy sheen. Cool before frosting, if using.
To frost: whisk powdered sugar with lemon juice and 2 teaspoons milk, adding more milk as needed for consistency. Drizzle over cookies and let set (it semi-sets, so hard on top but drippy underneath the top layer of icing).
Store in an airtight container for up to 3 days or freeze for up to 2 months.