Make the chimichurri: combine the parsley, oregano, and garlic in a food processor; pulse until coarsely chopped.
Add the paprika, vinegar, and oil; season to taste with salt and pepper; process until smooth.
Scrape into a bowl; let sit at room temperature for 30 minutes before serving.
The sauce can be made 8 hours in advance, tightly covered and refrigerated; bring to room temperature before serving.
Make the burgers: divide the meat into 4 equal portions; for each portion loosely into a 3/4 inch burger and make a deep depression in the center with your thumb.
Season both sides of each burger with salt and pepper.
Heat oil in a pan or griddle over high heat until the oil begins to shimmer; 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 until golden brown and slightly charred on the second side (4 minutes for med-rare beef, 5 minutes for turkey).
During last minute of cooking, cover each patty with a slice of cheese and cover with a basting cover or tent burgers with foil.
Place burgers on the bun bottoms and top with a large dollop of the sauce and sliced onion.
Cover with the bun tops and serve immediately.