Beat the butter in a mixing bowl. Add the salt, and then the sugars a little at a time, beating until creamy.
Add the egg and extracts, then beat on low speed until mixed.
Add the flour 1/2 cup at a time, mixing just until blended in. Do not overmix.
Divide the dough evenly between 3 bowls.
Add 25 drops of green food coloring to one of the bowls of cookie dough. Stir to distribute the color throughout the dough.
To another bowl of dough, add 25 drops of red gel coloring and stir until thoroughly mixed.
Cover the bowls with small plates. Stack the bowls on top of each other and refrigerate them for 4 hours to overnight to harden the dough.
Place a sheet of plastic wrap on a flat surface. Remove the bowl of green dough from the refrigerator. Use a spoon to remove spoonfuls of dough from the bowl. Form a dough log in the middle of the plastic wrap, 2 inches wide and about 1 inch tall. It should be about 8-10 inches long. First, form a rough log. Then use your hands to press and sculpt it until it is flat on the top.
Get the white dough from the refrigerator. Form another log of white dough in front of the green one, using the first log as a size guide. When they are close in shape and size, pull up on the plastic wrap to lift and flip the white log onto the green one. Adjust it as needed.
Finally, create a red log on the plastic wrap that is the same size as the other two. Use the plastic wrap to help lift it and position it on top of the white dough log. Further hand shape as needed.
Use a paper towel to wipe the plastic wrap clean from the red dough.
Preheat the oven to 350°F. Place a silicone mat or parchment paper on a baking sheet.
Cut the log into 1/4 inch pieces and place them 1 1/2 inches apart on the prepared baking sheet. Wipe the knife with a paper towel between cuts. Position the flags with the green stripe on the left side. Bake for 10-11 minutes, rotating the baking sheet after 5 minutes. Do not brown the cookies or they will lose their color.
Rewrap the dough and refrigerate while a batch of cookies bakes. Repeat until all the cookies are baked.