Place potatoes in a large skillet. Cover with cold water. Bring to a boil and simmer, covered, for 10 minutes. Set aside to cool.
Meanwhile, heat 2 tablespoons of the olive oil in a large nonstick skillet over medium heat. Add the onions and peppers to the pan, along with a big pinch of salt and black pepper. Saute for 10-15 minutes, or until tender and starting to brown. Remove to a separate bowl, leaving as much oil in the pan as possible.
Add 1 more tablespoon of olive oil to the pan and heat over medium heat. Saute the vegetarian sausage slices in the pan until browned on both sides, about 2-3 minutes per side. Remove to a paper towel-lined plate, leaving as much oil in the pan as possible.
Drain water from the potatoes and cut the potatoes in half. Heat the remaining tablespoon of olive oil in the skillet where you cooked the sausage and veggies, over medium heat. Add the potatoes and smoked paprika to the pan, along with a pinch of salt and black pepper. Saute until potatoes start to brown, about 6-8 minutes, stirring occasionally. Season to taste with salt and black pepper. Toss the potatoes with the onions, peppers, and vegetarian sausage.
In the same skillet, fry the eggs to desired degree of doneness. Serve potato and onion hash topped with one egg per person.
Adapted from BBC Good Food. I used Field Roast Mexican Chipotle Sausage Links for this, but pretty much any spicy vegetarian sausage will do!