In a food processor, combine goat cheese, sour cream, lemon juice, salt, pepper, and olive oil. Process until smooth. Place in a small container with a lid and refrigerate until ready to use.
In a large bowl combine lamb, chopped mushrooms, onions, parsley, garlic, salt, pepper spices, and chopped chile (or cayenne). With a fork, rake all the ingredients together until well combined. This method keeps the burgers from becoming to dense.
Heat the grill over medium-high heat.
Oil the grates and then place the burgers on the grill and cook for 6-8 minutes on each side.
Toast the buns a couple minutes on the grill.
Place a burger on the bottom of the brioche bun, then top with the goat cheese spread and some arugula, top with the bun and serve!