2 1/2 cups plus 6 tablespoons all-purpose flour Salt and freshly ground black pepper 1 teaspoon sugar 1 cup (2 sticks) plus 5 tablespoons unsalted butter, cut in small pieces 1 three- to four-pound chicken 4 cups , or canned low-sodium chicken broth, skimmed of fat Homemade Chicken Stock 8 ounces pearl onions 10 ounces red potatoes, scrubbed and cut into 1/2-inch pieces 5 medium carrots, peeled, cut into 1/2-inch pieces 14 ounces shiitake, cremini, or button mushrooms, quartered if large 3/4 cup milk 1 ten-ounce package frozen peas, thawed 2 tablespoons fresh thyme, leaves 2 tablespoons chopped fresh sage 1 large egg