Pat the duck breasts dry with paper towels. Score the skin in a crosshatch pattern 5 mm apart, cutting through the fat but not into the meat. Season both sides with the salt and pepper, and mince the shallot.

Set the breasts skin-side down in a cold 30 cm / 12-inch skillet and turn the heat to medium-low. Cook for 12 to 14 minutes, carefully pouring the fat into a heatproof jar every 4 to 5 minutes, until the skin is deep golden and crisp.

Flip the breasts and cook the flesh side for 3 to 4 minutes, until an instant-read thermometer in the thickest part reads 135 F / 57 C. Move the duck to a cutting board, skin-side up, and rest it for 10 minutes.

While the duck rests, pour off all but 1 tbsp of fat from the skillet and set it over medium heat. Cook the shallot for 2 minutes until soft and translucent, then add the thyme.

Add the balsamic vinegar, stock, and honey. Simmer for 5 to 6 minutes, scraping up the browned bits, until the sauce has reduced by about half and coats the back of a spoon. Take the pan off the heat, discard the thyme, swirl in the cold butter until glossy, and add a pinch of salt if needed.

Slice the duck across the grain into 5 mm slices and fan them on warm plates. Spoon the sauce over the meat, pour any board juices on top, and serve at once.