Preheat the oven to 400 degrees F. (see note for grilling instructions) Cut two sheets of heavy-duty aluminum foil large enough that they can be folded up into an 8-inch x 8-inch square in the middle.
In a large mixing bowl, toss the potatoes, carrots, onion, olive oil, and Italian seasoning together. Divide the veggies evenly between the two sheets of aluminum foil. (see note)
Combine the ground beef and salt in a medium mixing bowl. Divide into two 8-ounce portions and form them into patties. Place the burger patties in the center of the foil on top of the veggies.
Fold up the foil around the ingredients and seal well. Place on a large baking sheet with the burger patty side facing up. Bake for 30-35 minutes or until the burgers are fully cooked.
Carefully open the packets (there will be steam!) and top each burger with a slice of cheese. Place the baking sheet back in the oven, leaving the foil packets open, until the cheese has melted.
Remove from the oven and sprinkle with chopped parsley for garnish before serving warm. (If you prefer, you can transfer everything to plates. I like the easy cleanup of eating out of the foil packs.)
Notes
To more evenly divide the potatoes, you can prep them first. Add the potatoes, along with 1 tablespoon of olive oil and 3/4 teaspoon of Italian seasoning, to the large bowl and toss to coat. Divide the potatoes between the foil sheets. Then, toss the carrots and onions in the same bowl with 1 1/2 tablespoons of oil and 3/4 teaspoon of Italian seasoning, and divide them between the foil.
To Grill: Place the foil packets directly on the grill rack and cook over medium-high heat for 30-40 minutes, or until the potatoes are tender and the ground beef reaches an internal temperature of 160°F.