Remove burgers from fridge and allow to come to come room temperature, about 20 minutes.
In a small bowl, whisk together French onion mix and salt and pepper until well-incorporated; season burgers liberally on both sides, pressing seasoning lightly into burger.
In a heavy-duty saucepan with a lid, bring to medium-high heat, add vegetable oil, and cook burgers on one side until browned, about 3 to 4 minutes.
Flip burgers, add 1 heaping spoonful of French onions to each burger, and top with 1 to 2 slices of Gruyere cheese.
Pour beef broth into saucepan, place lid on saucepan, and continue to cook burger about 4 to 5 minutes, until cheese melts and burger has cooked through to desired doneness.
To toast buns, spread 1 tablespoon of mayonnaise on each side of bun and place in a separate pan; toast about 30 to 45 seconds until golden-brown.