Peel and slice the potatoes, carrots, and onion; cook in boiling, salted water to cover until very tender.
Mash through a wire strainer, saving the cooking liquid; or whirl in a blender with the liquid.
Melt the butter in a large pan, stir in flour until bubbly, and then pour in milk.
Cook, stirring constantly, until slightly thick.
Add the vegetable pulp and liquid.
Season with salt and pepper