Heat a griddle or large cast-iron pan over high heat. Add the canola oil and let heat until it begins to shimmer.
Divide the meat into 4 equal portions (about 6 ounces each). Shape each portion with your hands into a round patty, about 1 1/2 inches thick. Use your thumb to make a divot in the center of each burger. Sprinkle both sides of each burger with salt and pepper.
Put the burgers on the griddle or in the pan, divot-side up. Cook the burgers, flipping once, until golden brown on both sides and cooked to medium doneness, about 8 minutes total.
While the burgers cook, combine the tomato sauce, Calabrian chiles, basil and some salt and pepper in a small saucepan over medium heat. Cook until warm, 4 to 5 minutes, then remove from the heat and set aside.
Top each burger with 2 slices of the mozzarella. Carefully add 1/4 cup of water to the pan and immediately cover tightly with a lid. (If using a griddle, do one at a time and use a sandwich dome or stainless-steel bowl to cover.) Count to 10. Lift lid or dome to make sure that the cheese is totally melted (if not, cover again and count 5 second longer). Transfer the burgers to a plate and loosely tent with foil.
Place the burgers on the bottom halves of the focaccia slices. Top with the tomato sauce, arugula, Parmigiano-Reggiano, a drizzle of balsamic vinegar and the top focaccia slices and serve immediately.