Generously season the beef with salt and pepper and form into 2 (1/2-inch thick) oval patties.
Set aside, or place in the refrigerator while you cook the onions.
Add 1 tablespoon olive oil into a large nonstick skillet over medium-high heat. Once oil is hot, add onion and saute until browned, about 8 minutes.
Season the onions with red pepper flakes, salt, pepper, and chopped parsley. Remove onions from the skillet to a bowl and reserve.
Add the remaining tablespoon oil to the hot skillet. Once hot, add the burgers and cook for 3 to 4 minutes until very browned on one side. Flip and continue cooking the other side for another 3 minutes. Remove from pan.
Preheat a grill pan to medium-high heat.
Assemble the patty melts:
Top 2 pieces rye bread each with a slice of cheese, onions, the burger patty, a second slice of cheese, and a final piece of bread.
Butter both the top and bottom of the sandwich.
Grill each sandwich until golden and the cheese melts, about 3 to4 minutes on each side.
Serve hot with mustard and a side of pickles.
You want your patties thin and roughly the same size of the rye bread. Ground chuck is cut from the shoulder, ranges from 15 to 20 percent fat, and has a rich flavor that is tender and moist. This is best choice for burgers.