Heat oil in a 12-inch cast-iron skillet over medium-high heat. Add onion; sauté 10 minutes or until tender. Spoon onion into a bowl; let cool.
Add beef, pepper, and egg white to onion, and stir well. Divide beef mixture into 4 equal portions, shaping into 3/4-inch-thick patties.
Sprinkle salt evenly over surface of skillet. Heat salt in skillet over high heat 2 minutes. Add patties; cook 5 minutes on each side or until done. Place patties on bottom halves of rolls; top each with 1 teaspoon French dressing and top halves of buns.