Heat oven to 425°F; brush 18-by-13-inch rimmed metal sheet with some measured oil. Toss potatoes, bell pepper, and onion with remaining oil, paprika, black pepper, and 1/2 teaspoon salt.
Spread in one layer; roast 20 minutes. Turn; add cooked sausage.
Roast 10 minutes more until potatoes tender. Make 4 shallow wells exposing lightly oiled pan.
Crack 1 egg per well; add remaining salt. Scatter cheddar over potatoes, not yolks. Bake 7 to 10 minutes until whites and yolks set and eggs reach 160°F.
Lift each egg with potatoes using thin spatula; plate with 2 sausage links per portion.