For the caramelized onions: Heat the oil in a large cast-iron pan over medium heat, add the onions and cook, stirring occasionally, until soft and caramelized, about 10 minutes. Season with salt and pepper to taste, add the garlic and thyme and continue to cook for 3 minutes. Transfer to a plate and set aside. Reserve the cast-iron pan.
For the beef broth: Combine the beef broth, thyme and some salt and pepper in a medium saucepan over medium heat until the broth is warmed through and begins to take on the thyme flavor, 5 to 10 minutes. Remove the thyme and season with more salt and pepper to taste. Transfer to a bowl and top with more thyme leaves. Keep warm.
For the burgers: Divide the meat into 4 equal portions (about 6 ounces each). Shape each portion with your hands into a rectangular patty to match the shape of the hoagie rolls, about 3/4 inches thick. Sprinkle both sides of each burger with salt and pepper.
Place the hoagie rolls cut-side up on a baking sheet, brush the cut sides with some canola oil and sprinkle with salt and pepper. Toast in the oven until golden brown, 4 to 5 minutes.
Heat the same cast-iron pan over high heat. Add the canola oil and let heat until it begins to shimmer. Put the burgers in the pan and cook, flipping once, until golden brown on both sides and cooked to medium doneness, about 8 minutes total.
Top each burger with 2 slices of cheese. Carefully add 1/4 cup of water to the pan and immediately cover tightly with a lid. (If using a griddle, do one at a time and use a sandwich dome or stainless-steel bowl to cover.) Count to 10. Lift lid or dome to make sure that the cheese is totally melted (if not, cover again and count 5 seconds longer.)
Place the burgers on the bottom rolls and top with a large spoonful of the caramelized onions and the top rolls. Serve with the warm beef broth on the side for dipping.