Preheat oven to 400°F. Whisk together the flour, baking soda, and salt.
Beat the powdered sugar and butter until light and fluffy.
Add the egg and vanilla, and beat until thoroughly incorporated, about 30 seconds. Scrape down sides as necessary.
Slowly, mix the flour into the wet ingredients on low.
Flour the work surface and roll dough to about 1/4-inch thick, using flour for dusting as necessary.
Use the cookie cutters to cut the sugar cookies out, then transfer to baking sheet.
Bake the cookies for 7-8 minutes, then prepare the royal icing.