Beat butter and sugar with a stand mixer fitted with a paddle attachment on medium speed until light and fluffy, about 2 minutes. Add egg and vanilla extract; beat until fully incorporated, about 1 minute. Gradually add flour and salt, beating on low speed until combined, about 2 minutes. Shape dough into 2 (12- x 1-inch) logs.

Place sprinkles in a small, shallow, rimmed baking sheet. Roll each dough log in sprinkles, pressing gently to adhere. (Reserve any remaining sprinkles.) Wrap each log with plastic wrap; chill until firm, at least 2 hours or up to 2 days.

Preheat oven to 350°F. Line 3 baking sheets with parchment paper. Remove 1 dough log from refrigerator, and unwrap. If needed, press some of the reserved sprinkles onto log to fully coat. Slice log into ¼-inch-thick rounds (you will have about 48 rounds). Arrange about 18 cookie rounds spaced 1 inch apart on prepared baking sheet. (Place remaining cookie rounds on prepared baking sheets, and place in refrigerator until ready to bake.) Bake cookies in 3 batches until edges are light golden, 9 to 10 minutes per batch. Remove from oven; cool on baking sheets 5 minutes. Transfer cookies to wire racks, and cool completely, about 20 minutes. 

Meanwhile, remove second dough log from refrigerator, and repeat Step 3.