In a large mixing bowl, work the lemon zest into the sugar using the back of a spoon. Add the butter and cream until light and fluffy. Add the egg and vanilla extract and blend. Pour in the sourdough discard and blend into the wet ingredients until fully combined.
In a small bowl, combine the flour, salt, and rosemary, and whisk. Add the dry ingredients to the wet ingredients and gradually beat until fully combined.
Divide the dough in half. Roll out one half of the dough in the shape of a rough oval, to a 1/4 in thickness between 2 sheets of parchment paper. Repeat with the other half of the dough. Refrigerate for 2 hrs or up to 48 hrs.
Preheat the oven to 350. Line two large baking sheets with parchment paper. Take out the first half of dough and peel off the top parchment paper. Lay it down on your work station and dust the parchment paper with flour. Flip the dough over so the bottom is now on top and peel off the other sheet of parchment paper. Cut out your cookie shapes and place 1 in. apart on the cookie sheet. Sprinkle the tops with coarse sugar.
Bake for 10-12 min or until golden brown. Remove the cookies from the pan to wire racks to cool fully.