In the bowl of an electric mixer, combine the butter and sugar on medium-high speed until light and fluffy. Add the eggs and mix again. Scrape down the bowl and mix in the lemon juice and extract. Add the food color, if using.
In a separate bowl whisk together the flour, baking powder, lemon zest and salt. Add the flour mixture to the creamed mixture and mix on medium-low speed until well-combined. Cover and refrigerate the cookie dough in the mixing bowl for 30 minutes.
Preheat oven to 350°F.
Line two baking sheets with parchment paper. Place the coarse sugar in a bowl.
Portion the cookies by the heaping tablespoon (or cookie scoop) and roll each piece into a walnut-sized ball between your palms. Roll each ball in the coarse sugar. Place on the prepared cookie sheets about 2-inches apart. Bake for 15 minutes or until the cookies are puffed, crackled and light golden around the edges.
Let the cookies stand on the cookie sheet for five minutes before transferring to a wire rack to cool completely.
Swap the coarse sugar for powdered sugar, and bake as directed for lemon crinkle cookies.