Beat the butter, powdered sugar, granulated sugar and salt with an electric mixer on medium speed until pale and creamy, about 2 minutes. Add the egg and vanilla and mix on low to incorporate. Add the flour and beat on low speed until the flour is just incorporated. Divide the dough into 3 equal parts and place them in bowls. Add 15 drops green food coloring to one bowl and 25 drops of red to another and mix well to combine.
Lay a large piece of plastic wrap on a work surface. Starting with the green, pour the dough out into a long strip about 2 inches wide and 1 inch thick. Pour the white on top of the green in the same manner, forming another strip of the same dimensions; finish with the red. Wrap well, turn on its side so the colors are in the order of the Italian flag and refrigerate until firm, at least 3 hours.
Preheat the oven to 350 degrees F.
Line a baking sheet with parchment paper. Remove the plastic wrap from the dough and slice the dough into 1/4-inch slices. Place the cookies 1 inch apart on the baking sheet. Bake until just barely beginning to brown around the edges, 15 to 20 minutes. Cool for 5 minutes on the baking sheet before removing to a wire rack to cool completely. Repeat with the remaining dough.