Heat the canola oil in a large skillet over medium heat. Add the onion, carrot, celery, potatoes, sausage, paprika, garlic, red pepper flakes and 1/2 teaspoon salt. Saute, stirring frequently, until the potatoes and carrots are tender and the sausage is cooked through, about 15 minutes.
Meanwhile, bring a medium pot of water to a simmer over low heat and add the vinegar. Crack an egg into a small bowl and carefully slide it into the simmering water. Quickly repeat with the remaining eggs. Using a slotted spoon, carefully corral the white of each egg around its yolk. Poach the eggs until the whites are firm, or to the desired degree of doneness, 3 to 5 minutes.
Divide the hash among 4 plates. Top each with a poached egg. Season with salt and black pepper.
Photograph by Kang Kim
Courtesy of Food Network Magazine