Heat oven to 425° F. Rub the mushrooms with 2 tablespoons of the oil, and ¼ teaspoon each salt and pepper on a rimmed baking sheet.
Place the mushrooms stem-side down and roast until tender, 18 to 20 minutes. Top each mushroom with a slice of Cheddar and continue to cook until melted, 3 to 5 minutes more.
Meanwhile, on a separate rimmed baking sheet, toss the potatoes with the remaining 2 tablespoons of oil and ¼ teaspoon each salt and pepper. Roast, tossing once, until tender, 18 to 20 minutes; toss with the parsley.
Mix together the mayonnaise and mustard in a small bowl.
Stack the lettuce, mushrooms, onion, and mayonnaise mixture between the English muffins. Serve with the oven fries.