1Preheat grill to high heat. Brush the chicken breasts with the cooking oil and season liberally with salt and pepper. Grill 3 minutes per side, then reduce the heat on a gas grill to medium-low or move the chicken to indirect heat on a charcoal grill. Cover the grill until the chicken is cooked through, about 5 minutes.
2Meanwhile, in a saucepan, over medium-high heat, add jam, cognac, vinegar, ginger, soy sauce, salt, and blackberries. Mash have the blackberries with a potato masher or fork. Reduce for about 2 minutes while stirring.
3Slice chicken and transfer to a serving platter. Drizzle with the blackberry sauce and garnish with the chopped mint leaves. Pass remaining sauce at the table.