In a saucepan, heat butter, and add sliced onions. Cook and stir until the onions start to brown. Add balsamic vinegar, and cook until the onion is deeply caramelized. Season with salt and pepper. Set aside to cool.
In a bowl, combine ground lamb, garlic, mint, and cumin. Mix until just combined.
Shape the lamb into a ball. Form a well in the center, large enough for the goat cheese.
Place the cheese into the center of the ball, and enclose inside the ground lamb mixture. Gently form a patty. Season both sides with salt and pepper.
Cook patty in a hot cast-iron pan over medium-high heat for about 4-5 minutes each side.
Toast the brioche bun.
Spread mustard on the bottom and top bun. On the bottom of the bun place arugula, followed by lamb patty, a few spoonfuls of caramelized onion, tomato, and the top bun.