In a small bowl, using a fork, mash together the blue cheese and cream cheese until smooth. Stir in the parsley. Divide the mixture into 4 disks about 1 1/2 inches across. Cover tightly with plastic wrap and refrigerate for at least 20 minutes.
In a large bowl, season the beef with salt and pepper and mix well. Form into 4 equally sized patties about 3/4 inch thick.
Form a deep cavity in the center of each beef patty. Place a cheese disk in each cavity and pull the meat together to completely cover the cheese.
In a large skillet, heat the oil and butter over medium-high heat. Add the burgers and cook, about 4 minutes on each side for medium. Serve on the buns with the lettuce, onion and condiments.