For the burgers: Preheat the oven to 400 degrees F.
Place the bacon on a baking sheet lined with a rack. Roast until the bacon is crisp but still pliable, 20 minutes.
Using a large spoon, scrape the gills out of the mushrooms. Toss the mushrooms with olive oil and some salt and roast for 20 minutes.
Divide the ground beef into 8 portions; season generously with salt. Roll each portion into a ball and flatten into a patty. Make a nest or indentation in four of the patties and fill each with 2 ounces of the fontina. Place the remaining patties on top of the cheese-filled patties, pinch firmly to close, and then press and shape the patties.
Coat a large sauté pan with olive oil and add the onions. Season with salt and cook until the onions are starting to brown and are very soft and aromatic, 5 to 6 minutes. Remove from the heat.
Heat a large griddle or flat-top over medium-high heat. Coat the griddle lightly with olive oil and cook each burger until very brown and crusty, 4 to 5 minutes per side.
Cut the rolls almost all the way in half, leaving one side intact, so they open like a book. Place open-side down on the griddle and toast.
Place a burger on the bottom half of each roll. Break the bacon strips in half and place 4 halves on each burger. Divide the onions among the burgers. Cut each mushroom in 3 slices and arrange on top of the onions. Top with the remaining fontina. Place the burgers on the baking sheet and roast or broil until the cheese has melted. Top each burger with baby arugula and close the roll. Serve hot.
For the potato wedges: Toss together the Parmigiano-Reggiano, olive oil and potatoes in a bowl; season with salt. Lay the potatoes out in a single layer on a baking sheet. Roast for 10 minutes. Turn the potatoes over and roast for an additional 10 minutes.
Toss the potatoes with the parsley and serve alongside the burgers.