Peel and slice the onion. Grate the cheese. Peel the potatoes and cut them into thick slices (1/4 in/1/2 cm). Heat the oil and butter in a saucepan over a moderate heat, and fry the onions for 2-3 minutes, until soft. Add the raw vegetables (not the potatoes) and continue to fry gently for a few more minutes. Stir in the soup. Bring to the boil, and then lower the heat and simmer gently for 5-10 minutes, until the vegetables are tender, adding any cooked vegetables for the last 2-3 minutes to heat them through.
Meanwhile, partly cook the sliced potatoes separately in boiling, salted water for 4-5 minutes. Drain them. Arrange the mixed vegetables in a casserole or ovenproof dish. Stir in half the cheese and cover with the hot soup. Season with salt and pepper. Top with a thick layer of potato slices. Dot with some butter. Sprinkle with the rest of the cheese. Bake in a hot oven (400F/200C/Gas Mark 6-7) for 15-20 minutes, until the top is golden brown. Serve hot.