Sift flour, salt, and baking powder in a medium bowl.
In a large basin or stand mixer bowl with the paddle attachment, beat butter and sugar on medium speed for 2 minutes until light and fluffy.
While everything is being mixed, add the eggs one at a time. To clean the sides of the bowl, scrape them down. Mix in the vanilla extract.
Add the flour mixture slowly to the mixer and mix until fully combined. Split the dough in half, form two disks, and wrap in plastic. Refrigerate 2 hours or 2 days. (if chilling more than 2 hours, let dough remain at room temperature for 15–30 minutes to soften.)
Preheat the oven to 375F after chilling the dough.
Roll the dough to ¼-inch thickness on a lightly floured surface. Use 2½-inch cookie cutters to cut shapes. Roll again for more cookies.
Place cut-outs 1 inch apart on a non-stick or parchment-lined baking sheet.
Bake for 10–12 minutes until bottom edges brown. After a few minutes on the pan, transfer to a wire rack to finish cooling. Serving plain or with royal icing.