Preheat a grill to medium high. Heat the olive oil in a skillet over medium-high heat. Working in batches, add the salami in a single layer and cook, turning once, until crisp, about 1 minute per side. Drain on paper towels.
Add the mushrooms to the skillet and cook, undisturbed, until browned, about 2 minutes. Stir and continue to cook until tender, about 5 more minutes. Season with salt and pepper.
Form the beef into four 6-ounce patties, about 3/4 inch thick; press the center of each patty with your thumb to make a 1/2-inch-deep indentation. Season the patties on both sides with salt and pepper. Lightly brush the grill grates with vegetable oil, then grill the patties, indentation-side up, until marked on the bottom, about 5 minutes. Flip and cook until marked and slightly firm, about 3 more minutes for medium doneness. Remove the patties to a plate and let rest 5 minutes. Grill the rolls, cut-side down, until lightly toasted, about 1 minute.
Toss the arugula, vinegar, and olive oil to taste in a bowl; season with salt and pepper. Spread the ricotta on the bun bottoms and top with the burger patties, salami, mushrooms and arugula salad. Cover with the bun tops.
Photograph by Levi Brown
Courtesy of Food Network Magazine