Preheat oven to 350 degrees. Line a baking sheet with parchment paper.
In the bowl of a stand mixer fitted with the paddle attachment or a handheld electric mixer, cream butter until light and fluffy, about 2 minutes. Add the sugar, lemon juice, lemon zest and rosemary and beat at medium speed another 2 minutes. Reduce to low speed and add the flour into the wet ingredients. Scrape down the sides of the bowl. Form dough into a disc and wrap in plastic wrap. Chill for at least 1 hour.
Dump the dough out onto a floured surface and knead until smooth. Using a rolling pin, gently press on the dough and move slowly across the dough. Roll dough until it reaches 1/4 inch thickness. If the dough starts to break, add one teaspoon of water to make it more pliable. Using 2-3 inch round cookie cutters, cut out circles in the dough and place onto prepared cookie sheets.
Bake cookies 10-12 minutes, until they are slightly golden on top but still pale. Allow to cool on a wire rack before glazing.
For the glaze, whisk the confectioner’s sugar and lemon juice together in a small bowl. Pour the glaze into a decorator’s bag fitted with a Wilton #2 tip or into a Ziploc bag. If using a Ziploc bag, trim a tiny bit off the corner of the bag. Line cooled cookies up side by side and drizzle with the glaze in a zigzag pattern. Sprinkle with coarse sugar and allow to set for 30 minutes before serving.