Stir together ground lamb, garlic, mint, dill, salt, cumin, smoked paprika, cinnamon, and cayenne in a large bowl until just combined.
Shape lamb mixture into four 3-inch patties. Slice goat cheese log into 4 equal pieces. Place one goat cheese piece in the center of each patty and carefully wrap edges around cheese to enclose. Gently flatten each into a 4 inch patty.
Preheat an outdoor grill to high heat (450°F to 650°F) or a stovetop griddle to medium-high heat. Oil grate or griddle.
Grill patties until an instant-read thermometer inserted into centers registers 160 degrees F (71 degrees C) and goat cheese melts, 4 to 5 minutes per side.
Serve burgers on buns with arugula, aioli, and pickled red onions.