Preheat oven to 400°F. Combine butter, 3 tablespoons prosciutto, 2 1/2 teaspoons rosemary and 1 1/2 teaspoons mustard in bowl. Season with salt and pepper. Run fingers under skin over breast of each game hen, loosening skin from meat. Rub 1 tablespoon butter mixture under skin over breast of each hen. Sprinkle hens inside and out with salt and pepper. Place 1 teaspoons butter mixture in each cavity. Truss hens with kitchen string.
Place hens on rack in large roasting pan. Pour 1/3 cup broth over hens. Dot each hen with 1 teaspoon butter mixture. Roast hens 30 minutes, basting with remaining broth and butter every 10 minutes. Continue roasting without basting until juices run clear when thigh is pierced at thickest part, about 30 minutes more. Reserve 1/4 cup pan juices. Cool hens. Cut each hen in half.
Mix mayonnaise, apricot spread, remaining prosciutto, rosemary and mustard, and 1/4 cup pan juices in small bowl. (Hens and sauce can be made 1 day ahead. Cover and chill.)
Arrange watercress on large platter. Arrange hens atop. Serve with sauce.
