Prepare potatoes: Peel (optional) and dice potatoes into 1/2-inch (1–1.5 cm) cubes. Place diced potatoes in cold water and let sit 5 minutes, then drain and pat very dry with a towel.

Heat a large nonstick or cast-iron skillet over medium-high heat. Add 1 tablespoon oil and 1 teaspoon butter. When the butter foams and the oil shimmers, add the diced potatoes in a single layer. If your pan is small, cook in two batches to avoid crowding.

Cook undisturbed 4–5 minutes to develop a brown crust, then stir or toss to turn pieces. Reduce heat to medium and continue to cook, stirring every 2–3 minutes, until potatoes are tender and golden, about 8–10 more minutes total.

In the last 4 minutes of cooking, add the chopped onion and continue to cook until onion is translucent and slightly caramelized. Season with 1/2 teaspoon salt, 1/4 teaspoon black pepper, 1/2 teaspoon paprika, and 1/4 teaspoon garlic powder; toss to coat.

Taste and adjust seasoning. Sprinkle with 1 tablespoon chopped parsley if using. Transfer potatoes to a warm plate and keep warm while you finish the sandwich (cover loosely with foil).

Sandwich components

Cook the bacon: Place two strips of bacon in a cold skillet in a single layer. Turn heat to medium and cook, flipping occasionally, until crispy or to your desired doneness, about 8–10 minutes. Transfer bacon to a paper towel–lined plate to drain. Reserve 1 teaspoon bacon fat in the pan (or discard and use 1 tsp oil listed in ingredients).

Toast the bread: While bacon cooks (or after), toast two slices of your chosen bread (or split and toast an English muffin) until golden. Spread about 1 teaspoon butter or mayonnaise on the warm toast, if desired.

Fry the eggs

Heat a nonstick skillet over medium-low to medium heat. Add 1 teaspoon neutral oil or reserved bacon fat and swirl to coat. Crack two eggs into the pan, spacing them apart. Sprinkle with 1/8 teaspoon black pepper and a pinch (about 1/8 tsp) salt.

For sunny-side-up: Cook gently until whites are set but yolks are still runny, about 2–3 minutes; for over-easy: cook until whites are set, then carefully flip and cook 15–30 seconds more; for over-medium: 45–60 seconds more; for over-hard: 90 seconds to 2 minutes more. Remove eggs to a plate.

Melt cheese: Place one slice of cheddar or Swiss on one slice of hot toast, or in the skillet for 10–20 seconds until slightly melted. If using toaster and toast isn't hot enough, return to skillet briefly to melt cheese.

Assemble the sandwich: On the cheese-topped toast, place the lettuce leaf, tomato slices (season tomato with a small pinch of salt if desired), then the two bacon strips, and finally the two fried eggs. Top with the second slice of toast (buttered side down if you spread butter).

Serve immediately with the warm breakfast potatoes and optional condiments (ketchup for potatoes or hot sauce for eggs) on the side.