In a large bowl, season the ground beef with salt and pepper. Add 1/2 cup cold water and mix by hand until meat has absorbed all the water (the water keeps the burgers moist and juicy). Shape into 4 patties. Heat a large cast iron skillet over medium-high heat. Spray the skillet with non-stick cooking spray. Add the burgers and cook to desired temperature, about 8 to 10 minutes per side for medium-well. Remove the burgers from the pan and set aside. Add the onions slices to the same skillet and cook until golden brown on both sides, being careful to keep them in whole slices. It should take about 3 minutes per side. Remove from the pan and set aside.
Crack the eggs into the same skillet and cook until the yolk is not runny. Carefully remove from the pan. Lay the bottom of 4 hamburger buns on a flat surface. Top each bun with a burger. Top the burgers with 1 egg, onion slice, 1 slice of cheese and then the top bun. Lightly press the sandwiches together with your hand. Place the sandwiches into the pan and cook for 1 minute on each side to griddle the buns. Serve with French fries and pickles.