Brown the sausage 6 to 8 minutes: Heat a large 12 inch skillet over medium high. Add sausage and cook until browned with crisp edges and rendered fat. Break into bite size pieces. Transfer to a plate but leave 1 tablespoon fat in the pan. If dry, add 1 tablespoon oil.
Soften aromatics 3 to 4 minutes: Lower heat to medium. Add onion and bell pepper with a pinch of salt. Cook until lightly translucent and fond forms. This prevents raw bite later.
Load the potatoes 10 to 14 minutes total: Add remaining oil, then spread hash browns in an even 3/4 inch layer. Sprinkle salt, pepper, and smoked paprika. Press with a spatula for contact. Do not stir for 5 to 7 minutes until the bottom is deep golden and releases cleanly. If it sticks, it is not ready. Flip in big sections and cook 4 to 6 minutes more until crisp on the second side.
Bring it together 2 to 3 minutes: Fold in browned sausage. Taste and adjust salt. Sprinkle cheese over the top. Cover and let melt off heat 1 to 2 minutes. Top with green onions.
Optional eggs 3 to 5 minutes: Make 2 to 4 small wells. Crack in eggs. Cover and cook on medium low until whites set and yolks are jammy, about 3 to 5 minutes. For over easy, fry eggs in a second pan to keep the crust crisp.