How to Make It
Heat oven to 450° F. On a rimmed baking sheet, toss the carrots and rutabaga with 2 tablespoons oil, ½ teaspoon salt, and ¼ teaspoon pepper.
Roast the vegetables, tossing once, until golden brown and tender, 20 to 25 minutes.
Meanwhile, heat 2 tablespoons of the oil in a large skillet over medium-high heat. Add the shallots and cook, stirring often, until tender, 3 to 4 minutes. Stir in the vinegar; transfer to a bowl. Wipe out the skillet.
Form the beef into four ½-inch-thick patties and season with ½ teaspoon salt and ¼ teaspoon pepper.
Heat the remaining teaspoon oil in the skillet over medium-high heat. Cook the patties 3 to 4 minutes per side for medium-rare.
Serve the burgers on buns with the mayonnaise and lettuce, if desired. Top with the shallots and serve with the fries.