Preheat oven to 400°F (200°C). Arrange bacon slices in a single layer on a rimmed baking sheet lined with foil or parchment. Bake for 12–18 minutes, turning once if desired, until edges are crisp and centers are cooked through but not brittle. Remove bacon to a plate lined with paper towel to drain and keep warm.

Toast brioche

Split brioche buns and spread 1 teaspoon unsalted butter on each cut side (reserve remaining butter). Toast cut sides in a warm skillet or under a broiler until golden and lightly crisp, about 1–2 minutes per side in a skillet over medium heat or 1–2 minutes under broiler — watch carefully to avoid burning. Keep toasted buns warm.

Make mayo spread

In a small bowl, combine 4 tsp mayonnaise with 2 tbsp whole milk and a pinch (about 1/8 tsp) of salt and pepper. Whisk until smooth and slightly loosened. Set aside. (This makes a slightly creamy spread for the brioche.)

Cook eggs (folded/over-easy style)

Heat a 10–12 inch nonstick skillet over medium-low heat and add 1 tsp neutral oil and remaining 1 tsp butter. Crack two eggs into a small bowl and season with 1/2 tsp kosher salt and 1/4 tsp black pepper. Pour eggs into skillet and let cook gently until whites have mostly set, about 2–3 minutes. If you prefer folded/flat eggs, use a spatula to gently fold or flip once to cook the top lightly (over-easy) for about 20–30 seconds. Remove eggs to a plate. Repeat with remaining two eggs. Alternatively, for an even sandwich fit, you can scramble eggs lightly with the milk before cooking until softly set (about 1–2 minutes) — see note.

Assemble sandwiches

Spread about 1 teaspoon of the mayo mixture on the bottom half of each toasted brioche. Place one cooked egg on each bottom bun, then top each egg with two slices of bacon (arrange to cover the egg). Add a slice of cheddar cheese on top of the bacon while still warm so it softens. Sprinkle a little chopped chives or parsley over the cheese if using. Close with the top brioche half.

Finish and serve

If desired, return assembled sandwiches to the warm skillet, cover, and heat on low for 30–60 seconds to melt cheese and meld flavors. Serve immediately while warm. Provide extra salt and pepper to taste.

Notes & substitutions

For runnier yolks leave eggs over-easy and assemble immediately. For a firmer sandwich, scramble eggs with the milk before cooking and divide evenly between buns. Swap cheddar for gruyère or American cheese. Use pre-toasted brioche or a panini press for a crisper finish.