How to Make It
Using a sharp knife, score duck skin in a crosshatch pattern at 3/4-inch intervals, about 1/8 inch deep (being careful to avoid cutting into meat). Sprinkle duck breasts with 1 tablespoon salt, and chill, skin sides up, uncovered, 8 hours or overnight.
Place 15 charcoal briquettes in an even layer in a chimney starter; light briquettes. Once briquettes are covered with gray ash, use tongs to transfer 5 briquettes to one side of bottom grate of grill. Place 1 oak chunk on top of charcoal, and insert top grill grate. Cover and adjust vents to bring internal temperature to 85°F to 95°F.
Fill a rimmed baking sheet with a single layer of ice cubes. Place duck breasts, skin sides up, on a wire rack, and set rack over ice. Place baking sheet with duck on top grill grate over unlit side of grill. Cover and smoke duck, maintaining temperature of 85°F to 95°F (adding more briquettes if necessary to maintain temperature) and allowing duck breasts to become infused with smoke flavor, about 20 minutes. Remove duck breasts from grill, and set aside.
Heat duck fat in a large, high-sided skillet over medium-high. Add apples, and cook, stirring occasionally, until browned on all sides, about 6 minutes. Add butter, and swirl skillet to melt. Stir in sugar and remaining 1/2 tablespoon salt. Cook, stirring occasionally, until sugar melts and forms a golden caramel, 5 to 6 minutes. Reduce heat to medium, and carefully add duck stock. (Caramel will initially seize up and splatter.) Cook, stirring occasionally, until caramel is thick, syrupy, deep amber in color, and reduced to about 3/4 cup and apples are fork-tender, about 1 hour. Remove caramel from heat; turn off burner.
Place brandy in a heatproof measuring cup with a spout. Pour brandy over caramel. Using a long match or lighter, carefully ignite fumes just above surface of caramel. Allow flames to extinguish. Stir to combine. Keep caramel warm over low while preparing duck breasts.
Heat a large skillet over medium. Place duck breasts, skin sides down, in skillet, and reduce heat to medium-low. Cook, undisturbed, until fat is rendered and skin is golden brown, about 15 minutes, occasionally spooning off drippings from skillet. Increase heat to medium-high, and flip duck breasts. Cook until internal temperature registers 130°F for medium-rare, 6 to 9 minutes. Transfer duck breasts to a cutting board; let rest 10 minutes. Carefully add winter savory to skillet. (It will sizzle and splatter.) Cook until crisp, 10 to 20 seconds. Transfer to a plate lined with paper towels, and set aside.
Cut duck breasts into 1/2-inch-thick slices, and arrange on a serving platter with apples. Drizzle with caramel sauce, and garnish with crispy winter savory.