How to Make It
Heat a large skillet over medium-high heat. When hot, add the chicken sausage and cook for 5 to 6 minutes or until nicely browned. Remove from the pan and set aside.
In the same skillet, heat the olive oil over medium heat. When the oil is hot and shimmering, add the butternut squash, Brussels sprouts, and onion. Cook until very tender and lightly browned in spots, 18 to 20 minutes. (It’s cool if the squash starts to fall apart in the pan,dudes. It’s called hash for a reason.) Stir in the paprika and pepper and cook for 1 minute. Return the sausage to the pan and add the soy sauce. Taste the hash and season with a little kosher salt if necessary. Reduce the heat to medium-low.
Using a spatula (or the back of a spoon), make 6 small indentations in the hash. Crack the eggs into the indentations, cover with a lid, and cook for 8 to 10 minutes or until the whites have set but the yolks are still runny. (Keep in mind that the eggs will continue to cook once you remove them from the heat—be careful not to overcook them.)
Divide the hash among plates or bowls and serve with your finest hot sauce, if desired.