Preheat oven to 400°F (200°C). Line a rimmed baking sheet with foil and place a wire rack on top. Arrange 8 slices of thick-cut bacon in a single layer on the rack. Bake until crisp and evenly browned, 15–20 minutes depending on thickness. Remove from oven and transfer bacon to a plate lined with paper towel to drain. Keep oven on for optional warming of bread.

Prepare tomato

While bacon cooks, slice the ripe tomato into 1/4-inch (6 mm) thick slices. Lay slices on paper towel and season lightly with a pinch (about 1/8 tsp) of kosher salt and a few grinds of black pepper to draw out moisture and concentrate flavor. Set aside.

Make quick aioli (mayo spread)

In a small bowl, combine 3 tbs mayonnaise, 1 tsp Dijon mustard, 1 tsp apple cider vinegar, and 1 tsp olive oil. Whisk until smooth and slightly glossy. Taste and adjust with a pinch of salt and pepper if needed. This will be spread on the bread.

Pan-fry eggs

Heat a nonstick skillet over medium-low heat and add the 1 tbs softened unsalted butter. When butter is melted and foaming, crack 2 large eggs into the pan. Cook sunny-side up until whites are set but yolks remain runny, about 3–4 minutes. For over-easy eggs, cook 2–3 minutes then gently flip and cook 20–30 seconds more. Season eggs with a light pinch of kosher salt and black pepper while in the pan. Remove eggs to a plate and keep warm.

Toast bread

Lightly butter one side of each of the 4 slices of bread with the remaining softened butter (or use oven if preferred). Toast the bread, buttered-side down, in a clean skillet over medium heat until golden and crisp, about 2 minutes per side, or place in the oven at 400°F for 3–5 minutes until toasted. Remove and keep warm.

Assemble sandwiches

Spread about 1/2 tbs of the aioli mixture on the interior side of each slice of toasted bread (total roughly 3 tbs used across 4 slices). On two slices (the bottoms), layer: 2 butter lettuce leaves (or split 4 leaves between the two sandwiches), 4 slices of baked bacon, and 3 tomato slices (or enough to cover). Season tomatoes with a light sprinkle of salt and pepper. Carefully place one cooked egg on top of the tomato for each sandwich. Top with the remaining bread slices, aioli side down. Press gently to compact.

Finish and serve

If desired, use a sharp knife to cut each sandwich in half diagonally. Serve immediately while bacon and eggs are warm. Provide extra aioli or hot sauce at the table if desired.