Bring a large pot of water to a boil, salt it and cook the broccoli rabe until crisp-tender, 5 minutes. Drain, cool and finely chop.
Place the broccoli rabe, ground meat and pecorino-romano in a bowl. Season with the chile, garlic, onion, paprika, fennel seeds, salt and pepper. Add a healthy drizzle of EVOO. Mix together and form four 1/2-inch-thick patties.
Preheat a griddle or cast-iron pan over medium-high heat. Add the patties and cook, flipping once, for 6 to 8 minutes. Transfer to a plate. Lower the heat to medium and wipe the pan.
Brush the bread on one side with the butter. Build the patty melts with bread (buttered side out), giardiniera, provolone slice, patty, provolone slice and bread (buttered side out). Cook on the griddle, flipping once, until the cheese melts.