Put the potatoes in a medium saucepan of salted water. Bring to a boil, reduce the heat and cook until tender, 25 to 30 minutes. Drain.
Meanwhile, light a grill or preheat the oven to 500°. Brush the Portobellos with 2 tablespoons of the vegetable oil and sprinkle with 1/4 teaspoon of salt and 1/4 teaspoon of pepper. Grill the mushrooms or roast them on a baking sheet, turning once, for about 15 minutes, or until browned and cooked through.
Brush the hens with the remaining 2 tablespoons of vegetable oil and sprinkle them with 1/2 teaspoon of salt and 1/4 teaspoon of pepper. Grill over moderate heat or roast in the oven on a large rimmed baking sheet for 12 minutes. Turn and cook for about 12 minutes longer, or until the juices run clear when a thigh is pierced.
Peel the potatoes. Slice the potatoes and mushrooms 1/4 inch thick and transfer to a large bowl. Add the parsley, olive oil, garlic and vinegar and season with salt and pepper. Toss gently but thoroughly. Mound the salad on plates and top each serving with 1 of the hen halves.
Wine Recommendation: The clean taste of a Chardonnay-based Mâcon-Villages from Burgundy will highlight the Cornish hens and potatoes. Try the 1996 from Louis Jadot.