Sun-dried tomato aioli: Process all of the sun-dried tomato ingredients in a food processor, scraping the sides of the bowl down as needed until blended. This might take a few times before it gets nice and smooth.
Avocado mash: Combine ingredients in a bowl and mash with a fork until smooth.
Add butter to a hot skillet. Whisk 2 eggs in a bowl and then pour into pan in an even layer. Add shredded mozzarella to the middle and then fold in the sides of the eggs so that is nicely melts the cheese. Remove from pan.
Add bacon to same pan and cook on medium heat for about 8-10 min flipping halfway through. Bacon should be nice and crispy. Transfer to a paper towel lined plate.
Toast ciabatta and add sun-dried tomato aioli. Add eggs, bacon, and smashed avocado. Add a handful of arugula, giving it a nice squeeze of lemon overtop. Slice and serve. Enjoy!