Using a hand mixer or standing mixer, beat butter until creamy.
Add powdered sugar and beat until combined. Add the eggs, vanilla, and salt. Beat until well combined.
Gradually, add the flour until dough is moist and holds together, but doesn’t stick to your hands.
Place dough on a cutting board and form into a ball. Cut in half, then cover one half with plastic wrap and set aside.
Divide the other half into 6 pieces. The outer layers of the rainbow will need a little bit more dough, so make two of them slightly larger than the others.
Dye the dough with food coloring by placing a ¼ teaspoon of dye on the ball of dough and kneading it until it is one solid color. Gloves optional but suggested.
Starting with purple, roll the dough into a log about 6-8 inches (15-20 cm) long. This will be the center of the rainbow.
Next, roll out the blue dough with a rolling pin until it’s about ¼ inch (6 mm) thick, slightly longer than the purple, and wide enough to wrap around the purple dough. Repeat with the green, yellow, orange, and red doughs. The further away from the purple you get, the wider the dough needs to be to go around the previous color, making the red dough the widest.
Wrap the blue dough around the purple, pinching together any cracks that form, cutting off any excess dough. Repeat with the rest of the colors.
Wrap in plastic wrap and chill for 30 minutes.
Once chilled, remove the plastic wrap, and cut the dough in half to make 2 rainbow shapes. Set aside.
Cut the reserved dough in half, roll out to ¼-inch (6 mm) thickness, and make it slightly longer than the rainbow log.
Place the rainbow dough in the center and wrap the plain dough around it, again, pinching together any cracks and smoothing out the surface.
Repeat with the other halves. Wrap with plastic wrap and chill for another 30 minutes.
Preheat oven to 350˚F (180˚C).
Once chilled, remove the plastic wrap. Slice cookies about ½-inch (1 cm) thick and place on a parchment paper-lined baking sheet.
Bake for 20 minutes or until the bottom has browned slightly.