Preheat oven to 375 degrees F. Line a deep, 9-inch round pan with baking paper. Steam vegetables separately and drain well.
Mash vegetables either by hand, hand mixer or processor, each color in a separate bowl. Do not process too much, leave some pieces.
In a saucepan on a low setting, heat the oil and margarine, add the flour and mix into a roux, constantly stirring with a wooden spoon. Mix soup powder with water and add to the roux mixture; stir and remove from the heat.
In a bowl, mix together the mayonnaise, mustard and eggs, combining well. Return roux to low heat and add in the mayonnaise mixture. Continue stirring until thick with a wooden spoon. Do not allow roux to come to a boil. Divide roux into 3 and mix each third by hand with 1 of the 3 kinds of vegetables.
Layer the 3 mixtures in the prepared pan and bake uncovered for 15 minutes. Lower the heat to 350 degrees F and bake for an additional 30-40 minutes.