Heat oil in a large saucepan over medium heat. Add shallot and cook, stirring, until beginning to soften, about 1 minute. Add port and figs and cook, stirring, 1 minute. Add 1½ cups strawberries, 1 cup broth, 1 tablespoon vinegar, ¼ teaspoon salt and ¼ teaspoon pepper; bring to a boil. Reduce heat to a simmer and cook, stirring occasionally, until the strawberries are very soft and broken down and the sauce has reduced slightly, about 15 minutes.
Strain the sauce through a fine-mesh strainer into a 2-cup glass measuring cup or bowl (discard the solids). You should have about 1 cup liquid. (If you have more than that, return the sauce to the pan and continue reducing until you have about 1 cup.) Combine the remaining 2 tablespoons broth and cornstarch in a small bowl. Return the strained sauce to the pan along with the cornstarch mixture; cook over medium heat, stirring, until the sauce is thick enough to coat the back of a spoon and has reduced by half, 5 to 7 minutes. Combine 1 tablespoon of the sauce with the remaining 1 teaspoon vinegar in a small bowl to use as a basting sauce. Cover the remaining sauce to keep warm.
Meanwhile, preheat grill to medium.
Season duck breasts with the remaining ¼ teaspoon salt and pepper. Oil the grill rack (see Tip). Grill the duck, basting twice, until an instant-read thermometer inserted into the thickest part registers 150°F for medium, 4 to 8 minutes per side, depending on the thickness. Transfer to a clean cutting board and let rest for 5 minutes.
While the duck is resting, chop the remaining 1 cup strawberries. Slice the duck and serve with the sauce, garnished with the chopped strawberries and basil.
Make Ahead Tip: Cover and refrigerate the sauce (Steps 1-2) for up to 2 days.
Note: Port is a sweet fortified wine that provides depth of flavor in cooking. Look for it in your wine or liquor store.
Shopping tip: Boneless duck breasts range widely in weight, from about ½ to 1 pound, depending on the breed of duck. For this recipe, we recommend using smaller (about ½-pound) breasts—we prefer their milder flavor. Look for them near other poultry in the fresh or frozen specialty-meat section of large supermarkets or online at mapleleaffarms.com or dartagnan.com.
Tip: To oil a grill rack, oil a folded paper towel, hold it with tongs and rub it over the rack. (Do not use cooking spray on a hot grill.)