Prepare the duck breasts by trimming away extra fat, pin feathers, and silverskin. Slash the fat on the top in a criss-cross pattern, using a sharp knife, taking care not to cut into the meat itself.
Preheat the oven to 430°F (220°C). Make the gastrique base by combining the sugar and vinegar in a small saucepan. Bring to the boil over medium heat, swirling the mixture, and then lower the heat. Watch it carefully once you smell a caramel-y odor. Once the mixture becomes quite thick, add the kirsch or liqueur. Stir well with a whisk until you have a smooth mixture. Remove from heat.
In a sautoir or any oven-resistant saucepan, heat the olive oil on the stovetop over medium-high heat. Brown the duck breast, skin side down, for 3-4 minutes – the skin should look brown and crispy. Turn it over, then immediately put it into the oven. Roast for 10-12 minutes.
Remove the duck from the oven. Transfer the breast to a warm plate, and cover tightly with aluminum foil. Degrease the sautoir or saucepan using a spoon, leaving the meat juices, and place the pan back on the stovetop on medium heat. Deglaze with (add) the stock, bringing to the boil and scraping up the brown bits on the bottom. Add the gastrique base to this sauce.
If using fresh cherries, add half of them now to the sauce. If using frozen cherries, strain them, discarding the juice, and add half of them to the sauce as it boils down. Continue to boil for about 5-8 minutes.
When the sauce starts to thicken, taste and adjust the seasoning, adding salt and pepper as necessary. Add the rest of the cherries and continue to reduce until the sauce is syrupy.
Slice the duck into ½-inch slices (12 mm). If there are any juices left underneath, add them to the sauce.
Remove the sauce from the heat. If you like, whisk a bit of butter into the sauce, taste, and serve over the duck breasts. Voilà!