Chimichurri: combine parsley, oregano, and garlic in food processor until coarsely chopped. Add paprika, vinegar, and oil , season with salt and pepper, then process until smooth. Scrape into a bowl. Let sit at room temp for 30 min before serving. Sauce can be made 8 hrs in advance, tightly covered, and refrigerated. Bring to room temp before serving.
Divide meat into 4 equal portions. Form each portion loosely into 3/4-inch thick burger and make a deep depression in the center with your thumb.
Season both sides with salt and pepper.
Cook burgers, using the oil and topping each one with a slice of cheese and and a basting cover* during the last minute of cooking.
Place the burgers on the buns, top with a large dollop of sauce and sliced onion. Serve immediately.
*Basting Cover - an inexpensive aluminum dome used in diners. It looks like the lid of a sautee pan, except that it is domed to fit over the burger. If you don't have this, you can use a sturdy metal bowl, or tent each burger with a square of aluminum foil. The cover helps to properly melt the cheese.