Preheat the oven with a cooling rack-topped baking sheet on the top shelf of the oven to 425 degrees F. Heat a griddle pan or large cast iron skillet to medium-high heat.
Place the parsley and arugula in a food processor and pulse into a finely chopped mixture.
Prepare the steak by cutting it nearly in half horizontally and spread it open like a book.
Top the meat with the chopped salumi and the chopped greens, then return the food processor bowl (without rinsing) to the base. Top the greens layer with the cheeses, then tightly roll the meat into a long, thin log and tie it up with kitchen string to secure. Season the meat with salt and pepper and drizzle with extra-virgin olive oil. Brown the meat all over in the hot griddle pan, 5 to 6 minutes, and then transfer to the hot rack-topped baking sheet and finish cooking the steak in the oven, 10 to 12 minutes more for medium-rare to medium.
While the meat cooks, in a saucepot over medium-high heat, add the extra-virgin olive oil. Add 1/2 the garlic, chile, onion, and oregano and saute for 2 minutes. Stir in the tomato paste for 30 seconds, and then add the wine. Puree the tomatoes in the food processor and add to the sauce. Bring to a boil, and then reduce heat to simmer until thickened, 10 more minutes. Season with salt and pepper.
Melt the butter over low heat in a small saucepan with the remaining garlic. Toast the bread and brush liberally with garlic butter.
Slice the steak into thin wheels, discarding string, and arrange on the garlic toasts, topping with sauce and some torn basil. Serve a little pickled giardiniera salad along side in ramekins, as a garnish.