Preheat oven to 350°F (180°C). Line baking sheets with parchment paper.

In the bowl of a stand mixer, whisk together sugar, lemon zest, and rosemary by hand. Add butter; using the paddle attachment, beat at medium speed until fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add egg and vanilla, beating until combined.

In a medium bowl, whisk together flour, baking powder, and salt. With mixer on low speed, gradually add flour mixture to butter mixture, beating until a dough forms.

Divide dough in half. On a heavily floured surface, roll half of dough to ½-inch thickness. (Lightly flour top of dough if it sticks to rolling pin.) (Keep remaining dough covered with plastic wrap.) Using a 3-inch round fluted cutter, cut dough, rerolling scraps and cutting to use all dough. Using a small offset spatula, place 1 inch apart on prepared pans.

Bake, one pan at a time, until lightly browned and tops look dry, 10 to 12 minutes. Let cool on pans for 5 minutes. Using a large flat metal spatula, remove from pan, and let cool completely on wire racks.

Transfer colored Royal Icing to separate small pastry bags; cut a ⅛-inch opening in each tip.

Working with 1 cooled cookie at a time, dip cookie into reserved white Royal Icing, letting excess drip off. Place on a wire rack. Pipe green icing in alternating stripes to create a circle. (When the icing is the same consistency, it should melt together with the white icing.) Gently drag a wooden pick through center of colored icing to create a wreath design; immediately decorate with sprinkles as desired. Repeat with remaining cookies and icing. Let stand until icing is dry, at least 2 hours, or up to overnight. Store in an airtight container for up to 5 days.