Make the stuffing: Preheat oven to 350 degrees. Bake bread cubes in a single layer on a rimmed baking sheet until just golden, about 25 minutes. Let cool completely. Pulse in a food processor (in batches, if needed) until coarse crumbs form, about 15 seconds; transfer to a bowl. Add 1/4 cup oil; toss well. Spread breadcrumb mixture on the baking sheet, and bake, stirring occasionally, until deep golden brown, about 15 minutes. Transfer to a large mixing bowl.
Heat remaining 3 tablespoons oil and 2 tablespoons butter in a large skillet over medium-low heat until butter has melted. Add leeks, celery root, and garlic; cook, stirring occasionally, until leeks are soft, about 15 minutes. Raise heat to medium. Stir in mushrooms and thyme; season with salt and pepper. Cook, stirring occasionally, until mushrooms are soft, about 10 minutes. Remove from heat. Stir vegetable mixture into breadcrumb mixture; let cool completely.
Prepare the duck: Place duck breasts, skin sides up, on a cutting board. Using a sharp knife, score lines, 1/2 inch apart, in a cross-hatch pattern into the fat; cut almost all the way through, but do not cut into flesh. Season both sides with salt and pepper. Cut six 14-inch pieces of kitchen twine. Lay 3 pieces of twine about 2 inches apart on cutting board. Place 1 duck breast half, skin side down, crosswise over twine. Place about 1/2 cup stuffing on top of duck. Top with another breast half, skin side up. Tie together tightly with the twine. Repeat with remaining breasts. Stuffed duck can be refrigerated, covered, overnight. Bring to room temperature before cooking.
Bake remaining stuffing: Preheat oven to 375 degrees. Butter an 8-inch square baking dish; set aside. Stir together remaining stuffing and 11/4 cups stock in a large bowl. Transfer to prepared dish. Bake until top is crisp and pale golden brown but center is moist, about 30 minutes.
Meanwhile, cook the duck: Heat a large skillet over medium heat. Add duck breasts to skillet. Cook, turning once, until golden brown, 5 to 7 minutes per side. Transfer breasts to a plate; pour fat from skillet into a heatproof container. Return breasts to skillet, and place in oven. Cook until an instant-read thermometer inserted into the thickest part of the meat registers 125 degrees.for rare, 25 to 30 minutes. Transfer to a plate; let rest in a warm place, loosely covered with foil, 15 minutes. Pour fat from skillet.
Make the sauce: Lightly wipe out skillet, and place over medium-high heat. Add wine; cook until reduced by half, about 5 minutes. Add port, if desired, and stock. Simmer until reduced by half, about 5 minutes. Add accumulated juices from the duck to skillet. Season with salt and pepper; whisk in butter. Pour sauce through a fine sieve into a gravy boat. Slice duck, and serve drizzled with sauce, with baked stuffing on the side.
Chicken is also delicious in this recipe: Substitute four 12-ounce boneless skin-on chicken breast halves for the duck. In step 3, don't score the breasts. To cook, heat two teaspoons oil in a large skillet over medium heat; cook until golden brown, about 10 minutes. Flip chicken, and transfer to the oven. Cook until chicken reaches 175 degrees, about 25 minutes. For the sauce, simply substitute dry white wine for the red wine.