Heat olive oil in a large skillet over medium heat. Add the sliced onions and a pinch of salt. Cook, stirring occasionally, for 15 to 20 minutes until deeply golden and soft. The smell will be intoxicating. Do not rush this step. Patience transforms the onions into sweet, silky strands that are the soul of this dish. If they start to stick, add a splash of water or broth.

Push the onions to one side of the skillet. Add the ground beef and break it apart with a wooden spoon. Cook until browned, about 5 to 7 minutes. Stir everything together so the onions get cozy with the beef. The sizzling sound is the promise of something wonderful.

Sprinkle in the garlic powder, onion powder, paprika, and a few grinds of black pepper. Stir for 30 seconds until fragrant. Then pour in the beef broth, Worcestershire sauce, ketchup, and yellow mustard. Stir well. The mixture will look a bit loose, but that is perfect. It will thicken as it simmers.

Bring the mixture to a gentle bubble, then reduce heat to low. Let it simmer for 10 to 12 minutes, stirring occasionally. The sauce will reduce and coat the beef and onions in a glossy, savory glaze. Taste and adjust salt or pepper. This is the moment when the kitchen smells like a French bistro.

While the meat simmers, split your rolls and toast them under the broiler or in a dry skillet until golden. If using cheese, spoon the hot beef mixture onto the bun bottoms, top with shredded cheese, and broil for 1 to 2 minutes until bubbly and melted. Watch closely so they do not burn.

Close the sandwiches with the bun tops and serve hot. The first bite will be messy, warm, and utterly satisfying. Ethan always reaches for a second one before finishing his first.