In a large skillet or Dutch oven, melt 1 tbsp butter and 1 tbsp olive oil over medium-low heat. Add the thinly sliced yellow onions and a generous pinch of salt. Cook, stirring frequently, for 15-20 minutes until deep golden and caramelized. Adjust heat if needed to avoid burning.

Once onions are caramelized, push to one side. Increase heat slightly and add minced garlic and dried thyme to the center. Sauté for 60 seconds until fragrant, then stir into onions.

Increase heat to medium-high. Add ground beef to the skillet with onions. Break up meat and season with salt and pepper. Cook until browned completely.

Drain any excess fat if necessary, leaving some for flavor.

Sprinkle flour over mixture and stir for 1 minute. Pour in beef broth, Worcestershire sauce, and soy sauce, scraping up browned bits.

Bring to simmer and cook 5-8 minutes until thickened. Adjust seasoning.

Preheat broiler. Brush bun insides with butter or broth, toast 1-2 minutes until golden.

Spoon filling onto toasted bun bottoms. Top with cheese slices. Broil 1-2 minutes until cheese is melted and bubbly.

Garnish with chopped parsley if desired, top with bun halves, and serve immediately.