Combine flour, baking soda, and salt in a medium bowl. Set aside. In a large bowl, using a mixer, set on medium speed, beat sugar and butter until light and fluffy. Beat in eggs until well combined; stir in vanilla. Gradually stir in flour mixture until a soft dough forms. Shape dough into a ball and wrap in plastic wrap; chill for 2 hours.

Preheat oven to 375 degrees. Lightly grease 2 baking sheets. On a lightly floured surface, roll out dough to 1/8-inch thickness. Using an egg-shaped cookie cutter, cut out shapes and place on prepared baking sheets. Bake cookies until lightly browned around the edges, about 8 to 10 minutes. Cool cookies on sheets, slightly before removing to wire racks; cool completely.

Place colored sugars in a colorful pattern on a plate. Working with a few cookies at a time, cover cookies with vanilla frosting. Before the frosting sets, carefully invert each cookie and press into sugar design on plate. Repeat with remaining cookies, changing the design on the plate, if desired.