Using an electric mixer on medium speed, beat the butter and sugar together for about 3 minutes until creamy.
Add the eggs and vanilla, and mix until combined.
Add the flour, baking powder, and salt, and mix until smooth.
Place the dough on a clean surface, then divide into two equal portions. Shape each half into a flat disk and wrap tightly in plastic wrap. Chill for about 1 hour in the refrigerator.
Preheat the oven to 350°F (175°C). Line baking sheets with silicone liners.
Place the chilled dough on a floured surface. Roll the dough to 1/4- to 1/2-inch (0,25- to 0.5-cm) thick. (Thicker cookies will be softer.) Flour the dough and rolling pin as needed to prevent sticking.
Use cookie cutters to cut out your desired cookie shapes. Gather and re-roll the dough as needed to use all remaining dough. If the dough is cracking, let it sit for 5 to 10 minutes more before rolling.
Transfer the cut cookies to the prepared pans. Bake one pan at a time for 10 minutes, or until the edges are lightly browned. Then allow the cookies to cool completely before icing.
To make the royal icing:
Combine the confectioners' sugar, milk, corn syrup, and vanilla until smooth.
For colored icings, divide the icing into separate bowls and mix in gel food color to get your desired colors.
Transfer the icing into piping bags, then cut the tip of the piping bag with a pair of scissors just enough to allow the icing to flow with a little pressure on the bag.
Pipe the icing onto cookies and decorate with sprinkles.
Notes
*I like salted butter in these cookies to balance the sweetness of the icing, but you can certainly use unsalted if you prefer. Be sure to soften the butter properly so it's not too soft or warm. It should hold a thumbprint when pressed.
The yield will vary depending on the size of your cookie cutters.
Store in an airtight container at room temperature up to a week.