Combine green onion or white onion, bread crumbs, serrano peppers, chipotle peppers, and salt in a large mixing bowl. Add beef; mix well. Shape meat mixture into four 3/4-inch-thick patties.
For a charcoal grill, place patties on the grill rack directly over medium coals. Grill, uncovered, for 14 to 18 minutes or until an instant-read thermometer inserted into the side of a patty registers 160°F, turning once.
For a gas grill, preheat grill. Reduce heat to medium. Place patties on the grill rack; cover and grill as above.
Grill or toast buns or bread. Serve burgers on grilled or toasted buns or bread topped, if desired, with flavored catsup, tomato slices, red onion slices, and serrano peppers.