Heat the oil and butter in an 8- to 10-inch nonstick skillet until hot but not smoking. Add the onions and potatoes and cook, covered, for about 10 minutes, stirring occasionally to brown on all sides.
Add the tomato slices to the skillet, arranging them so they cover most of the potato and onion mixture. Cover and cook for 1 minute.
Meanwhile, break the eggs into a bowl. Add the chives, salt, and pepper and mix with a fork.
Preheat the broiler. Add the egg mixture to the skillet and stir gently with a fork for about 1 minute so the eggs flow between the potatoes and tomatoes. Slide the skillet under the broiler, 3 to 4 inches from the heat, and cook for about 3 minutes, until the eggs are set.
Slide onto a platter, cut into wedges, and serve.