In a mixing bowl, beat butter until creamy. Add the confectioners' sugar and beat until well mixed.
Mix in the flour, a little at a time, until it is incorporated into the dough. Add the rosemary and stir until distributed throughout the dough.
Turn the dough out onto a floured surface and roll it into a ball. Flatten the ball into a disc, wrap it in plastic wrap, and refrigerate for about 45 minutes or until chilled but not hardened.
Preheat the oven to 325°F. Then remove the dough from the refrigerator and make sure it is workable. If the dough isn't chilled enough, it can be difficult to move after cutting. If it is too cold, it is challenging to roll it flat. You want it somewhere in between. On a floured surface, roll the dough to 1/4 inch (about 6 mm).
Using a cookie cutter, cut out shapes and place them on a baking sheet lined with parchment paper or on a baking mat. You can also lightly grease the pan if not using paper or a mat. Using the cookie cutter, line it up exactly and place it over a cut-out shape. Then sprinkle the coarse sugar on the cookie. The cutter helps keep the sugar on the cookie rather than scattering across the baking sheet. Repeat for all the cookies.
Bake for 13-16 minutes or until very lightly browned and set. Let cool for a few minutes, and then move the cookies to a wire rack to cool completely.
Notes
Store Cookie Dough in the Refrigerator
Keep the dough in the refrigerator between batches of cookies to prevent it from getting too soft.
Baking Mat
A trick I use to cut out cookies is to roll out the dough on a baking mat. Then gently cut out the cookies on the mat, being careful not to cut the mat. Then remove the extra dough from around the cut cookies, leaving the cookies on the mat for baking. This helps ensure the cookies don't tear when you try to transfer them to the baking sheet after cutting them with the cookie cutter. You do not have to flour the baking mat if you roll the dough on it. You also will not need parchment paper with a baking mat.