Preheat the oven to 350 degrees F. Mix together eggs, milk, butter and mix until just combined.
Grease a 18×13 inch sheet pan and line with parchment paper. Pour the batter in the sheet pan and bake for 15-20 minutes, or just as the cake begins to brown. Let cool.
Use a 5-inch circle cutter to cut out 6 circles and remove from the rest of the sheet.
¼ teaspoon water
Add a ½ cup of frosting into six separate bowls. For each color, dissolve the dye in water and mix into the frostings until completely dyed.
Scoop each frosting into piping bags.
Assemble your cake by piping frosting from the outer edge and filling it in. Start with piping purple frosting on the bottom layer and carefully stacking another cake layer on top. On that layer, pipe blue frosting, starting from the outer edge and going inward, and stack a cake layer on top. Continue piping and layering until you complete your rainbow.
Crumble up some of the leftover cake and cluster it on top of your rainbow cake. Add some sprinkles, if desired. A little sprinkles never hurt anyone. 🙂