Season duck breasts generously with kosher salt and place in the fridge overnight, skin side up and uncovered.

Remove duck from the fridge 30 minutes before cooking to bring to room temperature. Score the skin in a crosshatch pattern and trim any excess fat (reserve this). Season again with salt and Chinese five spice on both sides.

Place duck breasts skin-side down in a cold skillet. Use a plate or grill weight to press them down gently.

Slowly raise the heat to medium-low, allowing the fat to render. Spoon off the fat as it releases and reserve in a heatproof bowl.

Cook skin side down for 4 minutes. Flip to the meat side for 30 seconds. Remove from the pan and let rest for 4 minutes.

Repeat the process—4 minutes fat-side down, 30 seconds meat side, 4 minutes rest—until the internal temp reaches 135°F for medium-rare. Let rest 5 more minutes before slicing.

For the sauce, return the pan to medium heat. Add 1 tablespoon butter and the blackberries. Stir and smash berries until softened.

Deglaze with cognac and cook for 2 minutes. Add chicken stock and demiglaze. Simmer for 2 more minutes, then transfer mixture to a blender and purée.

Strain through a fine mesh sieve back into the pan. Stir in balsamic vinegar, Dijon, and season to taste.

Slice duck into ½-inch pieces. Plate and spoon warm sauce over top. Serve immediately.