In a small bowl, combine the mayonnaise and hot sauce.
Spoon into a small plastic zip-top bag and set aside.
Heat the olive oil in a large non-stick skillet over medium heat.
Fry the eggs until sunny side up but not quite cooked all the way through, then transfer to a plate.
Lower the heat slightly, place four tortillas in the pan and divide cheese among them.
When the cheese is almost melted, top each with another tortilla.
Pour salsa over each quesadilla and flip immediately.
They should sizzle a little.
Cook for 30 seconds, then flip back over.
Top each with a fried egg and sprinkle with a little extra cheese.
Reduce heat to low and cook for another minute or two, or until cheese on top is melted and egg white is completely set.
Transfer quesadillas to plates, snip the tip off the plastic bag, squeeze on mayonnaise mixture in an attractive manner (zig-zags work nicely) and serve.