In a saucepan, melt the butter. Add the onion and cook over medium heat, stirring, until it begins to brown, about 10 minutes. Add the sugar, garlic, thyme and bay leaf and continue to cook until the onion is deep golden, about 20 minutes.
Add the broth, Worcestershire and vinegar and simmer, 30 minutes.
Meanwhile, make the burgers.
Divide the meat into 4 equal portions and shape into patties. Season with salt and pepper.
Heat the oil in a large cast-iron skillet over high heat. Once hot, add the burgers and cook, flipping once, until golden brown on both sides and cooked through, about 8 minutes total.
Top each burger with a slice of cheese and cover the skillet. Heat for about 1 minute, until the cheese melts.
To Assemble:
Remove the thyme sprigs and bay leaf from the au jus. Place the burgers on the buns along with creamy horseradish and arugula. Using a slotted spoon, remove some of the onions and top the burgers with them.
Serve the remaining au jus on the side for dipping.