In a large bowl of a stand mixer fitted with a paddle attachment, mix the butter and sugar on high speed for about 3 minutes until well combined.
1 1/4 cups (284g) salted butter, 1 1/4 cups (250g) granulated sugar
Add the egg, egg yolk, and vanilla and mix well, scraping down the sides of the bowl.
1 large egg, 1 large egg yolk, 1 teaspoon pure vanilla extract
Add in the flour, salt, and baking powder on low speed, mixing just until incorporated and the dough comes together. Shape the dough into a flat disk and wrap tightly in plastic wrap. Chill for 2 hours or up to 3-4 days in the refrigerator.
3 cups (423g) all-purpose flour, 3/4 teaspoon salt, 1/2 teaspoon baking powder
When ready to bake the cookies, heat the oven to 325°F. Remove the dough from the fridge and let it sit on the counter for 10 minutes to soften slightly.
On a lightly floured surface, roll out the cookie dough to 1/4-inch thick. Cut out using cookie cutters and bake on a baking sheet lined with parchment paper for 12 to 15 minutes, until the edges just barely start to turn golden. Cool completely before decorating with sugar cookie frosting or royal icing.