Heat the butter in a large skillet over medium heat. Once melted, stir in the onions with a big pinch of salt and pepper and the thyme. Cook for 10 to 12 minutes, stirring often, until the onions begin to soften. Turn the heat down to low and cook, stirring often, until the onions are dark brown and caramely, anywhere from 1 to 2 hours, or until the onions reach the color you desire.
This will take a while and the onions are apt to burn. Keep an eye on them and stir often, reducing the heat when needed. Add a splash of water to the pan if it gets too dry at any time.
To make the burgers, season the ground beef well with salt and pepper. Divide the beef into 4 or 6 patties - depending on how large you want them. It’s up to you! I prefer a thinner patty, so I usually make 6 thinner burger patties.
You can use the same skillet you cooked the onions in or use a new one. Heat the skillet over medium heat and add a small drizzle of olive oil if needed. Add the burger patties and cook until deeply golden. Flip the burger. Top with a few spoonfuls of the crumbled blue cheese (as much as you’d like!) and let the other side brown and the cheese melts.
Serve the burgers on the lettuce cups. Depending on the lettuce, I will use 1 or 2 pieces per burger. Top each burger with caramelized onions. Drizzle on the honey mustard and sprinkle on the chives. Serve!