To prepare ducks: Preheat oven to 350°F. Set aside necks and giblets for sauce. (Reserve livers for another use.) Pull off any fat. Rinse cavities with cold water and pat dry. Sprinkle ducks with salt and pepper inside and out; stuff with garlic cloves and thyme. With a skewer or fork, prick holes in the skin without piercing the flesh. (This allows fat to drain during roasting.) Tuck wing tips under and tie legs together with kitchen string, if desired. Set the ducks, breast-side up, on a rack in a large deep roasting pan.
Roast the ducks until a meat thermometer registers 180°F, the juices run clear when a thigh is pierced with a skewer, and legs move freely, 2 1/4 to 2 3/4 hours. After 1 hour, use a bulb baster to remove fat that has accumulated in the pan. Repeat every 30 minutes.
While the ducks roast, prepare stock & sauce: Heat 2 teaspoons oil in large heavy pot (4-quart capacity) over medium-high heat. Pat the necks and giblets dry and add to the pot; cook until browned, about 2 minutes per side. Transfer to a plate. Add onion and carrots to the pot; cook, stirring occasionally, until lightly browned, 5 to 6 minutes. Add broth, water, tomato paste, garlic, thyme, peppercorns and the reserved necks and giblets; bring to a simmer. Reduce heat to low, partially cover and simmer for 45 minutes. Pour the stock through a strainer into a medium saucepan, pressing on solids to extract maximum flavor. Set aside. (Refrigerate if holding for longer than 1 hour.)
When ducks are roasted, transfer to a carving board, cover loosely and let rest for 15 to 20 minutes before carving. Pour any fat and juices from the pan into a glass measuring cup and place in the freezer so the fat will rise to the top. Place the roasting pan over medium heat; add port and bring to a boil, scraping up any browned bits. Bring the reserved giblet stock to a simmer over medium heat. Strain the port from the roasting pan into the stock. Skim fat from the chilled pan juices; add the juices to the stock. Add dried cherries (or dried cranberries) and simmer for 1 minute. Add the cornstarch mixture and cook, whisking, until the sauce is slightly thickened, 1 to 2 minutes. Stir in vinegar and season with pepper.
To carve ducks: Use a sharp knife to separate legs from breasts at joints. Separate thighs from drumsticks at joints. Carve breasts into thin slices. Discard skin. Spoon sauce over each serving.