Start by making the Avocado butter. Place avocado and softened butter into a small food processor and blend until smooth and creamy. Add salt to taste and blend again, then set butter aside. It is possible to make the butter by hand, but for smoothest results I recommend using a blender or processor.
Make the BBQ Mayo by simply combining the BBQ sauce with mayonnaise. Set aside.
Prepare your ground beef by dividing equally into 8 parts, and forming into a ball. Do not over handle the meat or it will be tough.
Heatlarge cast iron skillet over medium-high with some spray oil. Split the muffins and place them face down on the warm pan/griddle to toast about 1-2 minutes, being careful that they do not burn. Remove and set aside. Spread the top half of each muffin with a generous amount of avocado butter.
Re-spray the pan with some oil, and gently crack each egg into it. You may need to work in batches so they all fit. Use a spatula or spoon to scrape together any running egg whites to form a nice tight egg circle, so it fits well within the muffin. Cook for 3-4 minutes then flip and cook a further minute on the other side for runny yolks. When cooked, place eggs onto a plate lined with paper towel until ready to assemble sandwich.
Spray a little more oil onto the pan, then place a beef ball on top. Use a very firm spatula to smash the beef patty against the griddle, to 1/4 inch thickness. Work quickly to repeat with all balls, and then season liberally with salt. After 3-4 minutes, a crust will form and the patties should be easy to release and flip. Turn them over and lay a slice of cheese on each one. Cook a further 3-4 minutes until the cheese has melted.
To assemble the sandwich: place a beef cheese patty onto the bottom bun, layer with the fried egg, drizzle with BBQ Mayo then replace the Avocado butter muffin top. Serve immediately.