Heat a 10-inch nonstick or cast-iron skillet over medium heat. Add butter and olive oil. When butter foams, add the diced onion and bell pepper and cook, stirring occasionally, until softened and translucent, about 4–5 minutes.

Brown sausage

Push the vegetables to one side of the pan and add the loose sausage to the open space. Break it up with a spatula and cook, stirring and turning, until no longer pink and edges are slightly browned, about 5–6 minutes. Stir the sausage and vegetables together. Add minced garlic and cook for 30 seconds until fragrant.

If the skillet becomes very dry or browned bits stick, add 1–2 tablespoons water to deglaze and scrape up browned bits. Taste a small piece of sausage and adjust seasoning if necessary.

Eggs & finishing

Whisk eggs

In a medium bowl, crack the eggs, add milk, salt, pepper, and smoked paprika (if using). Whisk vigorously until combined and slightly frothy, about 30 seconds.

Pour and scramble

Reduce heat to medium-low. Spread the sausage and vegetable mixture evenly across the pan. Pour the whisked eggs over the sausage and vegetables. Let the eggs sit undisturbed for 10–15 seconds, then gently stir with a spatula, pushing cooked edges toward the center so uncooked egg flows out. Continue folding every 15–20 seconds until eggs are mostly set but still moist, about 2–3 minutes total.

Add cheese

When eggs are almost done but still a little soft, sprinkle the grated cheddar evenly over the top. Cover the skillet with a lid for 30–60 seconds to melt the cheese and finish cooking the eggs gently. Remove from heat when eggs are set but still slightly creamy.

Taste and adjust seasoning with extra salt and pepper if needed. Garnish with chopped chives or parsley and serve immediately.