Make the Chipotle Aioli: In a small bowl, mix together the mayonnaise, minced chipotle pepper, adobo sauce, lime juice, and garlic powder. Stir until smooth and creamy, and then set aside.

Cook Sausage Patties: Heat a large skillet over medium-high heat, add the sausage patties, and cook for about 3-4 minutes on each side until browned and cooked through. Remove from skillet and keep warm.

Toast Buns: Spread butter on the cut sides of the brioche buns and place cut-side down in the skillet. Toast for about 1-2 minutes until golden brown.

Scramble Eggs: In a bowl, whisk together eggs, milk, salt, and pepper. Melt butter in the skillet over medium-low heat, pour the egg mixture in, and stir gently until soft and fluffy.

Assemble Sandwich: Spread a layer of the aioli on both halves of the toasted buns. Add a sausage patty, half of the scrambled eggs, and a slice of cheddar cheese. Top with the other bun and serve immediately.