In a large mixing bowl, beat the softened butter and sugar using an electric mixer, until creamy and smooth. Add the eggs and vanilla extract. Beat until combined. Don’t forget to scrape the bottom and the sides of the bowl to mix everything well.
In a separate bowl, whisk together the flour, baking powder, and salt. Add to the wet ingredients. Stir using a rubber spatula until combined and no lumps of flour remain.
Divide the dough into two pieces and form into discs. It will be easier for you to roll out one piece of the dough at a time. Cover or wrap with plastic wrap. Let the dough chill in the refrigerator overnight or for at least 1 hour.
Preheat the oven to 400ºF. Line two large baking sheets with parchment paper.
Remove the first piece of the dough from the refrigerator. Lightly flour the work surface and the rolling pin. Roll out the dough to about ¼-inch thick. Cut out the dough using a cookie cutter. Arrange the cookies on the prepared baking sheet leaving at least 1 inch of space between each cookie.
Bake in the preheated oven for 8-10 minutes or until the edges are slightly golden and set. Remove from the oven and let cool for 5 minutes on the baking sheet. Then, carefully transfer to a cooling rack.
Repeat with the remaining dough. Dust the work surface with flour again, if needed.
Serve the rolled sugar cookies with or without icing.
Prepare the icing. In a mixing bowl, whisk together the powdered sugar, milk, light corn syrup, vanilla extract, and a pinch of salt. If the mixture is too thick, you can add more milk until you reach the desired consistency. If the mixture is too thin, add more powdered sugar instead. Divide the icing into the small bowls and add food coloring.
Transfer each color of icing to piping bags (or just spread evenly over the cookies with a butter knife), and decorate your cookies. Then, let the icing harden and enjoy!