In a small bowl combine all ingredients for the sauce together, adjust the seasoning to taste, and transfer the sauce to a bowl. Set aside.
Heat olive oil in a skillet over medium heat.
Add ground beef, and season with salt and pepper to taste. Cook breaking the meat with a spoon until the beef is nicely cooked through. Keep it warm until needed.
Heat a bit of oil in a skillet over medium heat.
Add a tortilla and place 1 slice of American cheese, follow with some shredded cheese, and then with a spoonful of cooked hamburger meat.
Sprinkle some diced onions, and add the special sauce. Follow with more shredded cheese and 1 more slice of American cheese.
Fold the tortilla in half and cook until nicely crispy on both sides (about 1 minute per side).
Repeat with the remaining tortillas and ingredients to make 3 more quesadillas.
Slice each quesadilla into 2 triangles and serve with some lettuce, pickles, and more sauce on the side.
Notes
Be careful not to burn the tortillas. Cook each side for up to 1 or 2 minutes until golden brown or just enough for the cheese to melt
If you are making these beef quesadillas for a crowd then you can use the oven instead, this way you will make more in one go.
Leftovers can be stored in the fridge for up to 3 days nicely wrapped with cling film and placed in an airtight container.