Step 1

Set giblets from duck aside; reserve liver for another use. Trim skin from both openings of the cavity; set aside for rendering. Cut both leg-thigh pieces off duck. Cut wings from breasts. Carve breasts from carcass for 2 boneless breasts. Trim off wing tips and flats; set aside. Cut carcass into quarters. Cover and chill leg, breast, and wing pieces.

Step 2

Heat reserved duck skin in a large heavy pot over medium heat. Cook, stirring, until about 2 Tbsp. fat is rendered; discard skin. Add giblets, wing tips and flats, carcass pieces, onion, and thyme sprigs. Cook, stirring occasionally, until lightly browned, about 15 minutes. Add celery, carrot, peppercorns, and 6 cups water. Bring to a boil over high heat, skimming any foam that rises to the surface. Reduce heat to medium; simmer duck stock for 1 hour.

Step 3

Meanwhile, using a sharp knife, cut all peel and white pith from 1 orange; set peel aside. Working over a medium bowl, cut between membranes to release segments into bowl; set aside. Add orange peel to simmering stock. Juice remaining 2 oranges and set juice aside.

Step 4

Strain stock through a fine-mesh sieve into a medium bowl (you should have about 2 cups). Transfer orange peel to a work surface; discard remaining solids in sieve. Slice peel into thin strips (remove white pith for a more refined look, if desired); set aside. DO AHEAD Stock can be made 1 day ahead. Cover and chill stock and peel separately.

Step 5

Prick duck pieces all over with a sharp knife. Season with salt and pepper. Arrange them in a single layer in a large heavy skillet (use 2 if needed), skin side down, and cook over medium heat, occasionally pouring off fat from skillet into a heatproof bowl, until skin is deep golden brown, 18–20 minutes. (Cover and chill fat; save for cooking potatoes or stir-fried rice.)

Step 6

Turn over duck pieces. Cook breasts, skin side up, until an instant-read thermometer inserted horizontally into thickest part of breast registers 125° for medium-rare, about 2 minutes. Transfer breasts to a large rimmed baking sheet. Continue cooking legs and wings until well browned on both sides and juices run clear when thigh is pierced with a fork, 10–15 minutes longer. Transfer to sheet with breasts. Pour off fat from skillet, leaving browned bits behind.

Step 7

Preheat oven to 450°. Return skillet with browned bits to medium-high heat. Pour in Port and cook, scraping pan with a wooden spoon (the browned bits will add extra flavor to the sauce), until reduced and syrupy, about 2 minutes. Add reserved orange juice and duck stock; boil, stirring occasionally, until sauce is smooth and thickened, about 20 minutes. Season sauce to taste with salt and pepper. Cover and keep warm.

Step 8

Place rimmed baking sheet with duck pieces in oven. Bake until duck is just heated through, 5–8 minutes.

Step 9

Pour warm sauce onto a deep platter. Scatter orange segments and strips of peel over. Top with duck pieces.