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.