In a medium bowl, whisk together 4 cups all-purpose flour, 1 teaspoon baking powder, and ½ teaspoon salt; set aside.
In a small saucepan, combine 1 cup (2 sticks) salted butter, 1 cup granulated sugar, and ⅔ cup light corn syrup. Cook and stir over medium heat until the butter is melted and the sugar is dissolved. Do not overheat.
Pour the butter mixture into a large mixing bowl. Stir in 1 tablespoon vanilla extract. Let the mixture cool about 20 minutes, until it is warm to the touch but not hot (so it will not cook the egg).
Add the 1 large beaten egg to the cooled mixture and mix until well combined.
Add the flour mixture to the wet mixture and stir until a cohesive dough forms.
Divide the dough in half, wrap each half in plastic wrap, and flatten each into a disk. Chill for at least 2 hours. (If chilled overnight, let the dough stand 5 to 10 minutes at room temperature before rolling.)
On a lightly floured surface, roll one dough disk to 1/8-inch thickness. Keep the other disk chilled until ready to roll.
Cut shapes with floured cookie cutters or a sharp knife. Place the cutouts 1 inch apart on an ungreased cookie sheet.
Preheat the oven to 375°F (if not already preheated). Bake the cookies 8 to 10 minutes, or until the edges are lightly browned.
Cool the cookies on the cookie sheet for 1 minute, then transfer them to wire racks to cool completely.