Ingredients:

• 2 teaspoons olive oil
• 1 medium-size (11-ounce) yellow onion, thinly sliced (about 2 cups)
• 1/4 teaspoon pure cane sugar
• 1 to 3 teaspoons water (if needed)
• 1 medium garlic clove, grated
• 1/4 cup mayonnaise
• 2 teaspoons fresh lemon juice
• 1/4 teaspoon kosher salt
• 1/4 teaspoon ground black pepper

• 8 thick-cut pepper bacon slices (about 12 ounces)
• 4 brioche sandwich buns, split
• 2 tablespoons unsalted butter
• 1 small (6-ounce) green bell pepper, finely chopped (about 3/4 cup)
• 10 large large eggs, beaten
• 4 ounces Gruyère cheese, shredded (about 1 cup)
• 3/4 teaspoon flaky sea salt, plus more for sprinkling
• 1/4 teaspoon cracked black pepper, plus more for sprinkling

Cooking Instructions:

Make the caramelized onion aioli:

1. Heat oil in a large nonstick skillet over medium-low. Add onion and sugar to skillet; cook, stirring occasionally, until onion is golden brown and softened, about 25 minutes. If onion slices stick to skillet and are beginning to burn, stir in water, 1 teaspoon at a time, to release onion slices from skillet. Stir in garlic; cook, stirring constantly, until mixture is fragrant and flavors meld, about 2 minutes. Remove from heat; spread onion mixture in an even layer on a large heatproof plate. Refrigerate, uncovered, until cool, about 15 minutes. Roughly chop onion slices. Stir together onion, mayonnaise, lemon juice, kosher salt, and ground pepper in a small bowl until well combined. Set aside.

Make the sandwiches:

1. Arrange bacon slices in a single layer in a large nonstick skillet. Cook over medium, turning occasionally, until crisp and rendered, about 10 minutes. Transfer bacon to a paper towel–lined plate, reserving drippings in skillet. Working in 2 batches, toast brioche buns, cut side down, in drippings in skillet over medium until golden brown, about 1 minute per batch. Transfer buns to a large plate, and set aside. (Do not wipe skillet clean.)
2. Add butter to skillet, and heat over medium until melted. Add bell pepper; cook, stirring occasionally, until softened but still bright green, about 2 minutes. Add eggs; cook, stirring often, until half of egg mixture consists of small curds, about 2 minutes. Stir in cheese; cook, stirring often, until all eggs form small, soft curds and cheese is fully incorporated, 1 minute to 1 minute and 30 seconds. Sprinkle with flaky salt and cracked pepper. Remove from heat.
3. Spread 1 tablespoon caramelized onion aioli on each bun half. Top bottom bun halves evenly with egg mixture. Break bacon slices in half, and place 4 pieces on each egg pile. Top with remaining bun halves. Sprinkle tops of buns with flaky salt and cracked pepper. Serve immediately.