Melt butter in a large skillet over medium heat. Add thinly sliced onions and cook, stirring every few minutes, for 12 to 15 minutes until deeply golden brown and fully caramelized. If onions stick, add a small splash of water to deglaze and continue cooking.

Push onions to one side of the pan. Add the ground beef to the cleared space and cook, breaking it up, until browned and fully crumbled with no pink remaining. Drain any excess grease.

Stir the onions back through the beef. Add minced garlic and dried thyme. Cook for 60 seconds, stirring constantly, until fragrant.

Sprinkle flour evenly over the meat and onion mixture. Stir continuously for 1 to 2 minutes to cook out the raw flour taste.

Pour in the beef broth and Worcestershire sauce. Stir well to incorporate. Bring to a simmer and cook for 5 to 7 minutes, stirring occasionally, until the sauce has thickened into a velvety gravy that coats the beef. Season with salt and pepper.

Split and lightly toast the buns. Spoon the beef mixture generously onto the bottom halves and top each with one or two slices of Swiss cheese.

Place the open-faced sandwiches on a baking sheet and broil for 1 to 2 minutes until the cheese is fully melted and beginning to bubble and brown at the edges. Watch closely.

Top with the remaining bun halves. Garnish with fresh parsley if desired and serve immediately.