Preheat the oven: Preheat your oven to 300°F.
Prepare the eggs: In a bowl, whisk together 8 large eggs, ½ cup whole milk, 1 teaspoon hot sauce, and a pinch of salt and pepper. Grease an 8×8-inch baking dish with butter (or spray oil). Pour the egg mixture into the greased dish.
Create a water bath: Fill a sheet pan with hot water, then place the 8×8-inch baking dish on top of the water. Carefully place the sheet pan in the oven and bake at 300°F for 40 to 45 minutes until the eggs are fully set.
Make the sweet and spicy bacon: While the eggs are baking, prepare the bacon. In a small bowl, mix together 2 tablespoons brown sugar, 2 tablespoons maple syrup, ½ teaspoon red chili flakes, and ½ teaspoon black pepper. Spread the mixture evenly over the 8 slices of thick-cut bacon. Place the bacon on a separate sheet pan and bake in the oven for the same amount of time (40-45 minutes) until crispy and caramelized.
Broil the eggs: Once the eggs are fully set, remove them from the oven. Top the egg dish with 4 slices of cheese and broil for 1 minute, just until the cheese gets melted and gooey. Cut the eggs into 4 equal pieces.
Toast the brioche buns: Butter and toast 4 brioche buns until golden.
Make the spicy mayo: In a small bowl, mix together ½ cup mayonnaise, 1 tablespoon Sriracha (or chipotle sauce), 1 teaspoon lime juice, and a pinch of salt and pepper. Adjust to taste.
Assemble the sandwiches: Spread a generous layer of the spicy mayo on each toasted brioche bun. Add a small handful of arugula on top of the mayo to help it stick. Place one of the egg patties on each bun and top with two slices of the sweet and spicy bacon. Serve hot and enjoy your souffle egg sandwich!