Chop the salmon fillets into 1 - 2-inch chunks. Place a quarter of it into a food processor bowl with 1 tablespoon of the mayonnaise and the red wine vinegar, and pulse it into a paste. Add the onion, garlic, bell pepper, and the rest of the salmon. Pulse slowly/carefully, scraping down the sides as needed, until everything is just barely chopped finely and mixed well with the first salmon paste. (If needed you can chop the salmon entirely by hand.)
Turn mixture into a large bowl and add the bread crumbs, Parmigiano-Reggiano, dried basil, salt, pepper, and cayenne. (I stirred those ingredients together in a cup first.) Mix gently until only combined; don’t overmix. Divide salmon mixture into four even piles. Shape each into a ball and gently pass the ball back and forth between your hands to firm it up. Pat the balls into even round patties, tapping the edges to make them even. Cover and refrigerate at least 20 minutes, but no more than a few hours.
Heat the butter and oil in a large nonstick skillet or sauté pan over medium high heat. Add burgers and cook 4 minutes until lightly browned. Turn carefully, add 1/4 cup grated mozzarella (or 1 slice) to each patty. Lower heat and cook another 2 minutes or so or until an instant read thermometer reads around 140 degrees F. (The temperature will rise as the burgers rest.) Don’t overcook.
Serve hot or warm on buttered, grilled buns with mustard sauce, and/or the reserved 2 tablespoons mayonnaise (I like to mix the sauce and the mayo), fresh basil, and sliced tomatoes.LEFTOVERS: Store up to 3 days well-wrapped in refrigerator. Eat cold on salad or reheat gently in greased pan. Freeze well-wrapped for up to 3 months; thaw overnight in fridge; reheat gently in greased pan.