Put the wine, port, ginger, and sugar (1 Tbs. for sweet cherries; 2 Tbs. for sour) in a medium saucepan over high heat. Bring to a boil, reduce the heat as needed, and simmer until reduced by half, 8 to 12 minutes.
In a small bowl, mix the balsamic vinegar and cornstarch. Whisking constantly, add the cornstarch mixture to the wine mixture and simmer until slightly thickened, about 1 minute more. Discard the ginger. Reduce the heat to low, add the cherries, and simmer slowly until the cherries are warm, 1 to 2 additional minutes. Season to taste with salt and pepper. Set aside.
Put the duck breasts on a work surface skin side up. Using a sharp knife, score the skin in a grid pattern, cutting just halfway through the thickness of the skin. Season the duck generously with salt. Scatter the cracked pepper on a small baking sheet and press the duck into the pepper, coating both sides evenly.
Heat a heavy-duty 12-inch skillet over medium-high heat. Arrange the duck skin side down and cook until the skin is deep golden-brown and the fat from the skin has rendered, 6 to 8 minutes. Carefully spoon off and discard all but about 1 Tbs. of the fat. Turn the duck and continue to cook until an instant-read thermometer inserted in the thickest part of a breast reads 135°F for medium rare, 4 to 8 minutes. Transfer to a cutting board, cover loosely with foil, and let rest for 10 minutes.
Meanwhile, warm the sauce over very low heat. Cut each duck breast on the diagonal into 1/2-inch-thick slices and serve with the sauce.