Prepare the Cookies: Beat butter and sugar with a heavy-duty stand mixer fitted with a paddle attachment on medium speed until light and fluffy, 3 minutes. Add egg and vanilla; beat until incorporated, 30 seconds. Gradually add flour and salt, beating until combined, 1 minute. Remove dough from bowl; shape into a flat disk. Cover with plastic wrap; chill at least 2 hours or up to 24 hours.

Preheat oven to 350°F. Line 2 baking sheets with parchment paper. Transfer dough to a lightly floured work surface; roll to ⅛-inch thickness. Using a 4 ½-inch dove-shaped cookie cutter, cut out dough; reroll scraps twice to cut 36 Cookies total. Arrange 2 inches apart on prepared baking sheets. Flip half of the Cookies to face opposite direction. Bake in 2 batches until slightly golden on edges but pale in centers, 8 to 10 minutes. Cool on baking sheets 5 minutes. Transfer to a wire rack; cool completely, about 30 minutes.

Prepare the Royal Icing: Beat powdered sugar, meringue powder, and 6 tablespoons of the warm water with an electric mixer on low speed until combined, 1 minute. Increase speed to medium, and beat until smooth, about 2 minutes. Beat in remaining 2 tablespoons warm water, ¼ teaspoon at a time, as needed until desired consistency is reached. Transfer 2 tablespoons Royal Icing to a small bowl; stir in 1 drop food coloring until well combined. Cover and set aside. Spoon remaining undyed icing into a ziplock plastic freezer bag with a small corner snipped off (or into a piping bag fitted with a small round tip).

Pipe a border of undyed icing around each Cookie. Flood with additional icing. Using a wooden pick, spread icing to piped border inside edges to fully ice Cookies. Let stand until icing is set, 1 hour. Pipe wings onto Cookies with some of the undyed icing. Spoon reserved green icing into a ziplock plastic freezer bag with a small corner snipped off (or into a piping bag fitted with a small round tip). Pipe 1 olive branch beneath each dove’s beak. Let stand until icing is set, 30 minutes.

Place 2 (6-inch) lollipop sticks or wooden skewers on a sheet of parchment paper. Pipe a ½-inch-long strip of undyed icing along the top of each stick. Place 1 Cookie on top of icing on each stick; press lightly to adhere. Let stand until dry, 2 hours. Serve remaining Cookies with the cake, or store in an airtight container up to 3 days.