Start by placing bacon on a parchment-lined baking sheet. Cook in a cold oven set to 425°F for about 20 minutes until crispy. Drain on paper towels to remove excess grease, keeping it flavorful but not oily.
Blend avocado, basil, garlic, lime, lemon, jalapeño, and salt in a food processor until smooth. Add 1-2 teaspoons of water if the aioli is too thick. This sauce adds a creamy, zesty layer to your sandwich.
Rub chopped rosemary onto the hash browns and cook in a skillet with a drizzle of oil until golden brown, about 4-5 minutes per side. Cook eggs in the same skillet to your preference—fried, scrambled, or over-easy work beautifully.
Toast English muffins cut side down in the skillet for 30-45 seconds until golden. Top one half with cheddar and cover for 30 seconds to melt. The other half gets the aioli, ready to assemble.
Layer the aioli-topped muffin with arugula, hash browns, eggs, bacon, and the cheese-topped muffin. Slice in half and enjoy immediately for the perfect balance of flavors and textures.