Louisiana Burger

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1/2 cup prepared mayonnaise

1 tablespoon Dijon mustard

1 tablespoon whole-grain mustard

1 teaspoon hot sauce, or more if desired

5 cornichons, diced

1 green onion, white and pale green part, finely chopped

1 tablespoon finely chopped flat-leaf parsley

Salt and freshly ground black pepper

1 tablespoon sweet paprika

2 teaspoons freshly ground black pepper

2 teaspoons kosher salt

1 teaspoon dried thyme

1/2 teaspoon garlic powder

1/2 teaspoon onion powder

1/2 teaspoon cayenne pepper

1 1/2 pounds ground chuck

2 tablespoons canola oil, plus more for brushing

8 paper thin slices pepper jack cheese

4 buns, split

8 slices thinly sliced tasso ham

1 red onion, peeled and cut into 1/4-inch thick slices