In the mixing bowl of a stand mixer, add the butter, sugar, and egg, and blend with the whisk attachment until smooth, and creamy. Add the vanilla, and mix until blended.
In a separate medium bowl, add the flour, and salt, and mix well to blend. Add the dry ingredients to the mixing bowl, and mix until blended, and smooth.
Place a large piece of parchment paper on the kitchen counter, and sprinkle with flour. Place half of the cookie dough on the parchment paper, and with a floured rolling pin, roll the dough to 1/2 inch thickness.
Cut out the donuts with the cookie cutter, and place on a parchment paper lined cookie sheet. Bake at 350 degrees for 10 to 12 minutes, or until cookies being to turn golden brown around the edges. Remove from oven to a wire rack, and cool completely.
To make the royal icing
In the mixing bowl of a stand mixer, place the warm water, meringue powder, and Karo syrup, and blend at medium speed using the whisk attachment until completely blended and smooth.
Add the powdered sugar, about 2 cups at a time, and mix on low, until all the powdered sugar has been added. Scrape down the sides of the mixing bowl as needed, and blend until the icing is smooth and creamy.
In two bowls, place about 1/3 of the Icing in each bowl, and leave the remaining icing in the mixing bowl. In one of the bowl, add several drops of blue food coloring gel, until you have the desired color of icing. In the other bowl, add a couple drops of red food coloring gel, and stir well. If needed, add more red food coloring gel, until you have the desired color.
Place the icing in a plastic pastry bag, (you'll have three bags, one red, one white, and one blue), and snip off the very end of the bags. Pipe the icing around the outside of the cookies, and pipe around the inside of the donut hole.
Fill in with the Icing
Add sprinkles of choice. Allow about 1 hour for Icing to completely set.