Heat the oven to 425°F and brush an 18-by-13-inch rimmed metal sheet pan with a little of the measured oil. Toss potatoes, pepper and onion with remaining oil, paprika, pepper and 1/2 teaspoon salt.
Spread vegetables in one layer and roast for 20 minutes, then turn them and add the cooked sausage links.
Roast for 10 minutes more until potatoes are tender. Push vegetables aside to make 4 shallow wells exposing the lightly oiled pan.
Crack 1 egg into each well, sprinkle with remaining salt and scatter cheddar over the potatoes rather than the yolks. Bake for 7 to 10 minutes until the whites and yolks are set and eggs reach 160°F.
Use a thin spatula to lift each egg with some potatoes onto a plate. Serve with 2 sausage links per portion.