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

Toast the Buns: Melt a tablespoon of butter in a skillet over medium heat. Toast the brioche buns cut-side down for 1-2 minutes until golden brown.

Cook the Eggs: In the same skillet, add butter or oil if needed, then crack the eggs in and cook to your desired doneness. Add pepper jack cheese on top to melt.

Assemble the Sandwiches: Spread chipotle aioli on toasted buns, layer arugula, cheesy egg, and avocado, then close the sandwich.