Add the flour, powdered sugar, and salt to a large mixing bowl and mix to combine. Add the cold butter, cut into small cubes, and the egg. Mix everything until a smooth dough forms.
Knead everything together with your hands just until a smooth dough forms, being careful not to over-knead it, as this can make the cookies tough. Once the dough is ready, refrigerate it for about 30 minutes.
Preheat the oven to 350°F (180°C).
Lightly flour your work surface and roll out the chilled dough to about 1/4 inch (5–6 mm) thick. Using your favorite cookie cutters, cut out different shapes from the dough.
Carefully transfer the cookies to a parchment-lined baking sheet, leaving a little space between each one.
Gather any leftover dough, gently bring it back together, and roll it out again to cut more cookies.
Place the cookies on a parchment-lined baking sheet and bake for about 10 minutes, or until the edges are lightly golden.
While the cookies are in the oven, prepare the icing. Combine the icing sugar and the remaining ingredients in a large bowl. Beat on low speed first, then increase to medium-high and mix for about 1 minute, until the icing is smooth, thick, and glossy.
Divide the icing between small bowls, depending on how many colors you want to make. Add a few drops of food coloring to each bowl and mix well. If the icing becomes too runny, just mix in a little more powdered sugar to thicken it.