Preheat oven to 350°F. Line 3 baking sheets with parchment paper. Beat granulated sugar, 1 cup of the softened butter, and 2 teaspoons of the vanilla with a stand mixer fitted with a paddle attachment on medium speed until fluffy, about 1 minute. Add egg, beating until blended, about 30 seconds.

Stir together flour and salt in a small bowl; gradually add to butter mixture, beating on medium speed until just combined. Add rainbow sprinkles, beating until just combined.

Divide dough into 2 equal portions; flatten each portion into a 1-inch-thick disk. Wrap each disk in plastic wrap; chill 15 minutes.

Unwrap 1 of the dough disks, and place on a heavily floured surface. Roll disk to 1/8-inch thickness. Cut out rounds from disk using a 2 1/2-inch round cutter. Place rounds 2 inches apart on prepared baking sheets. Repeat process with remaining dough disk. Reroll dough scraps, and cut out rounds from dough. Discard remaining dough scraps.

Bake cookies in preheated oven in batches, 1 baking sheet at a time, until cookies are lightly browned, 8 to 10 minutes per batch. Let cookies cool on baking sheets 5 minutes. Transfer cookies to wire racks, and let cool completely, about 15 minutes.

Beat remaining 1/2 cup softened butter with an electric mixer on medium speed until creamy, about 1 minute. Gradually add powdered sugar and remaining 1 teaspoon vanilla, beating until combined. Gradually add milk 1 teaspoon at a time, if needed, until smooth and creamy, about 30 seconds.

Working with 1 cookie at a time, spread 2 to 3 teaspoons frosting on cookie, and immediately garnish with sprinkles.