Heat a griddle or cast iron pan over medium-high heat.
Combine the meat, chile pepper, garlic, rosemary, parsley, salt, pepper, and a little extra-virgin olive oil. Form into 6 thin patties. Drizzle the patties with a little more oil and fry on the hot griddle for 3 to 4 minutes per side. Remove from the heat.
Lightly butter the bread. Build the patty melts, layering the bread (buttered sides facing out) with 1 slice of cheese, a few leaves of arugula, the patty, a second slice of cheese, and finally another slice of bread. Cook the patties on the griddle to melt the cheese, 2 minutes on each side. Cut the patty melts from corner to corner and serve immediately.