Heat oven to 450° F. On a rimmed baking sheet, toss the potatoes, 2 tablespoons of the oil, ¾ teaspoon salt, and ¼ teaspoon pepper.
Roast, tossing once, until golden brown and tender, 22 to 25 minutes.
Meanwhile, in a medium bowl, toss the fennel, onion, lemon juice, 1 tablespoon of the oil, and ¼ teaspoon each salt and pepper. Set aside.
Form the beef into four ½-inch-thick patties. Season with ½ teaspoon salt and ¼ teaspoon pepper. Heat the remaining teaspoon of oil in a large skillet over medium-high heat.
Cook the patties for 3 to 4 minutes per side for medium-rare. During the last minute of cooking, top them with the cheese and cook, covered.
Place the burgers on the buns with the fennel relish and arugula, if desired. Serve with the fries.