Heat a large skillet over medium heat. Add the bacon strips in a single layer and cook, turning occasionally, until crisp, about 8–10 minutes. Transfer cooked bacon to a paper towel-lined plate to drain. Reserve 1 tsp bacon fat in the pan and discard the rest (or save for another use).

Cook peameal

In the same large skillet, add 1 tablespoon vegetable oil and 1 tablespoon butter over medium-high heat. When hot, add the peameal slices in a single layer (work in batches if necessary). Cook 2–3 minutes per side, until golden brown and heated through. Remove to a plate and keep warm. Wipe skillet if excess fat accumulates, leaving a thin film for flavor.

Toast buns

Halve the buns or English muffins. Melt the remaining 2 tablespoons butter in a separate large pan or in the oven on a baking sheet. Brush or spread butter on cut sides and toast in a skillet cut-side down over medium heat until golden, about 1–2 minutes, or broil 1–2 minutes in the oven. Keep warm.

Fry eggs

In a nonstick skillet over medium heat, add a small pat of butter or a light spray. Crack eggs into the pan and cook sunny-side up until whites are set but yolks still runny, about 2–3 minutes. If you prefer over-easy, flip gently and cook an additional 15–30 seconds. Season each egg with a little salt and black pepper.

Assemble sandwiches

Spread 1 tsp mustard and 1 tsp mayonnaise on each bottom bun. Place a lettuce leaf on the bottom (if using). Add one hot peameal slice per sandwich, then top each peameal with a slice of cheddar while still hot so it softens. Place a bacon strip (or folded two strips to fit) on top of the cheese, then place the fried egg on top of bacon. Add 2 pickle slices if using. Finish with top bun and press gently.

Serve

Serve immediately while warm so the cheese is melty and the egg yolk is runny. Provide extra mustard and ketchup at the table if desired.