In a small bowl, combine ½ cup mayonnaise, 1 to 2 minced chipotle peppers in adobo sauce, 1 tablespoon of freshly squeezed lime juice, and 1 clove of minced garlic. Stir the mixture until it's smooth and well blended. Set this zesty chipotle aioli aside to let the flavors meld as you prepare the other components of your Spicy Chipotle Egg Breakfast Sandwiches.

Slice the brioche buns in half and heat a skillet over medium heat. Add 1 tablespoon of butter, allowing it to melt and sizzle. Once hot, place the cut-side of the buns down in the skillet and toast for 1-2 minutes, or until they are golden brown and crispy. Remove the buns from the skillet and set them aside, ready for assembly.

In the same skillet, add another tablespoon of butter and allow it to melt. Crack 2 to 4 large eggs into the skillet, cooking them sunny-side up for about 2-3 minutes, or until the whites are set but the yolks remain runny. In the last minute of cooking, top the eggs with pepper jack cheese, covering them briefly until the cheese melts beautifully, adding a creaminess to your Spicy Chipotle Egg Breakfast Sandwiches.

Spread a generous layer of your chipotle aioli on the toasted side of each bun. Begin layering on the sandwich by placing a handful of fresh arugula, followed by the cheesy eggs, and then slices of creamy avocado. Finish by topping with the other half of the bun, creating a delightful, hearty sandwich, perfect for any morning or brunch occasion.