MAKE THE FISH MIXTURE: Add the first ten ingredients to the bowl of a food processor (cod-cayenne pepper). Pulse slowly/carefully, scraping down the sides as needed, until everything is just barely mixed and holding together. (Alternately, chop fish finely and mix in a large bowl with remaining ingredients.) Do not overmix. If the mixture isn’t holding together well, add another tablespoon mayonnaise.
FORM THE PATTIES/REFRIGERATE: Turn the fish mixture out onto a cutting board and divide 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. Don’t overwork them as they'll be tough +/or come apart easily. Cover and refrigerate at least 20 minutes, but no more than a few hours, to firm them up.
COOK THE PATTIES: Heat the butter and oil in a large nonstick skillet or sauté pan over medium high heat. Gently add fish patties and cook 4 minutes until lightly browned. Turn carefully, add sliced Pepper Jack and sliced Cheddar, if using. Lower heat, and cook another 2-3 minutes or so or until cheese is melted, patties are browned/cooked through, and an instant read thermometer reads 140+ degrees F. (The temperature will rise as the burgers rest.) Don’t overcook.
SERVE HOT on buttered, grilled buns with extra Pepper Jack, if using; tartar sauce; lettuce; tomato; and pickled onions.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.