Combine the remaining orange sauce mixture with the duck breasts in a resealable plastic bag or airtight container. Refrigerate for at least 2 hours and preferably overnight. (Alternatively, if you’re pressed for time, you can skip the marinating and go straight to searing the duck since the final sauce contributes most of the orange flavor to the duck. Go ahead and use all the orange sauce mixture in the final orange sauce—you’ll simply have more orange sauce to drizzle, dunk, and douse your duck breast.)
Preheat the oven to 300°F (150°C)
Remove the duck breasts from the marinade, discarding the marinade. Pat the duck breasts dry with paper towels. Using a paring knife, score the skin by making slices 1/4 inch (6 mm) deep, spacing them about 1/2 inch (12 mm) apart. Be careful not to cut into the meat beneath the skin and fat.
Place the breasts, skin side down, in an ovenproof sauté pan or cast-iron skillet that’s large enough to contain both duck breasts without crowding them. Place the pan over low heat and cook the duck for about 15 minutes, allowing the fat to render, or melt, very slowly and pouring off the fat as it accumulates in the pan. (You can reserve the fat for another use. We suggest you sizzle or roast up some potatoes after you toss them in the rendered duck fat.)
Transfer the pan with the duck breasts to the oven and cook until an instant-read thermometer inserted in the thickest part of the duck breast registers 135°F (57°C) for medium-rare or 140°F (60°C) for medium. You'll want to check the duck after 10 minute but it could take as long as 25 minutes. When the duck is done, transfer it to a wire rack placed over a rimmed baking sheet, tent the duck breasts loosely with foil, and let rest for at least 5 minutes.
Carefully pour off the excess fat from the pan used to cook the duck and return the pan to medium-high heat. When the pan is hot, toss in the shallot, jalapeño, and scallions and cook, stirring occasionally, until browned and softened, about 3 minutes. Meanwhile, in a small bowl, whisk the cornstarch into 1/4 cup chicken stock. Add the orange mixture, the cornstarch mixture, and the remaining 1/4 cup chicken stock to the pan, bring to a simmer, and cook, stirring until warmed through and lightly thickened, 1 to 2 minutes. Taste and adjust the seasoning as desired. Remove from the heat and cover to keep warm.
To serve, thinly slice the duck breasts against the grain and arrange on a platter. Pour the warm sauce over the duck. Serve with steamed rice on the side.