You can grind as much or as little meat together as you'd like, using equal parts of each type.
Cut both meats into small pieces. Grind through the large holes of a meat grinder. Combine in a bowl, then grind together through small holes of grinder. Divide into 4 equal parts (each about 5 1/2 ounces). Form into patties, each about 3 1/2 inches in diameter.
Preheat grill to medium-high. Skewer onion slices crosswise with presoaked skewers; brush both sides with oil. Season both sides of burgers with salt. Grill burgers 2 1/2 minutes a side for medium-rare, placing cheese on burgers in last minute. Transfer to a plate.
Grill onion until tender and lightly charred, about 4 minutes a side. Meanwhile, grill buns until toasted, 30 seconds. Sandwich burgers and onions (removed from skewers) on buns with mustard mixture, lettuce, and tomatoes.