4 portobello mushrooms (about 1 1/4 pounds total), stems discarded
1/4 cup olive oil
kosher salt and black pepper
4 slices Cheddar (about 4 ounces)
3 medium russet potatoes (about 1 1/2 pounds), cut into wedges
2 tablespoons chopped fresh flat-leaf parsley leaves
1/3 cup mayonnaise
2 tablespoons Dijon mustard
lettuce and sliced red onion, for serving
4 English muffins, split and toasted
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.