Preheat oven to 400°F (205°C). Line a rimmed baking sheet with foil and place a wire rack on top. Arrange bacon slices in a single layer on the rack. Bake 15–20 minutes, turning once if needed, until crisp and evenly browned. Remove and transfer to a paper towel–lined plate to drain. Keep warm in a low oven (about 200°F/95°C) if assembling later.

Prepare buns

While bacon cooks, halve the brioche buns. Heat a large skillet over medium heat and add 2 tablespoons butter. When butter foams, place cut sides of buns in the skillet and toast until golden, 1–2 minutes. Remove and set aside toasted sides up.

Mix spread

In a small bowl combine mayonnaise (6 tbs) and Dijon mustard (1 tsp). Season with a pinch (about 1/8 tsp) of salt and pepper. Mix until smooth. If using avocado, mash half of it into the spread for an avocado-mayo variation and reserve the other half for slices.

Slice tomato and prep lettuce

Slice tomatoes into 1/4-inch thick slices (about 3–4 slices per tomato). Rinse and pat dry butter lettuce leaves. Season tomato slices lightly with a pinch of kosher salt (from the 1 tsp) and a little black pepper.

Fry eggs

In a nonstick skillet, heat 1 tablespoon neutral oil over medium. Crack eggs into the pan and cook sunny-side up until whites are set but yolks are still runny, about 2–3 minutes; if you prefer, cover pan for 30–45 seconds to finish whites. Season eggs with remaining kosher salt and black pepper. Alternatively cook over-easy if you want slightly runny but sealed yolks.

Melt cheddar over eggs (optional)

If you prefer melted cheddar on the egg, when eggs are nearly done, place one cheddar slice over each egg and cover the pan for 15–30 seconds until cheese softens.

Assemble sandwiches

Spread about 1.5 tablespoons of the mayonnaise mixture on the bottom half of each toasted bun and 1 tablespoon on the top half. Layer 2–3 butter lettuce leaves on each bottom bun, then 2–3 tomato slices. Top tomatoes with 3 bacon slices per sandwich, arranging to cover the tomato. Place the cheddar-topped egg on the bacon. If using avocado, add 3–4 thin slices on top of the egg. Cap with the top bun.

Serve

Press gently to compact, slice in half if desired, and serve immediately while warm. Optionally add extra black pepper or hot sauce to taste.