Bring a pot of water to a boil and salt it. Blanch the broccoli rabe for 5 minutes, drain and cool. Finely chop and reserve.
Place ground meat in a bowl and season with salt, pepper, fennel, chili, garlic, onion and juice, paprika, Romano, chopped rabe and a healthy drizzle of EVOO, twice around the bowl. Mix and form 4 large, thin patties about a half-inch thick.
Heat a griddle or cast-iron pan over medium-high heat. Cook patties 3-4 minutes on each side. Remove to plate. Reduce heat to medium. Wipe pan clean.
Brush bread on one side with melted butter. Build patty melts: a slice of bread with buttered side facing out, some relish, 1 slice of cheese, patty, another slice of cheese and bread with the buttered side facing up. Grill patty melts until deeply golden on both sides and cheese has melted. Cut patty melts corner to corner and serve.