Divide the meat into 4 equal portions (~6 oz. each); form each portion loosely into a ¾ inch thick patty and make a deep depression in the center with your thumb.
Season both sides of each burger with salt and pepper.
Brush the patties with the oil.
Heat a grill pan over high heat on top of stove; cook the burgers until golden brown and slightly charred on the first side, about 3 minutes for beef, 5 minutes for turkey.
Flip burgers over; cook beef burgers until golden brown and slightly charred on second side, 4 minutes for medium rare or until cooked to desired doneness (cook turkey burgers until cooked throughout, about 5 minutes on second side).
During last minute of cooking top each patty with a slice of Swiss cheese and cover with a basting cover, inverted sturdy metal bowl, or tent patties with foil.
Add the tomato slices to the grill pan, cover each with a slice of American, a slice of goat cheese, and some Parmesan; cover and melt for 1 minute.
Arrange some arugula leaves on the bun bottoms and top each with a burger and a tomato slice.