Preheat oven to 250 degrees F (120 degrees C). Line a 9x13 baking dish with 2 pieces of aluminum foil.

Stir kosher salt, 2 teaspoons black pepper, and dried thyme together in a bowl. Mix garlic, ginger, orange zest strips, fresh thyme, and bay leaves together in another bowl.

Season duck all over, inside and out, with about 2/3 of the kosher salt mixture. Fill duck cavity with garlic mixture.

Place duck, breast-side up, into prepared baking dish and season with remaining salt mixture. Wrap duck tightly in aluminum foil.

Roast in the preheated oven until meat pulls away from the bones, 5 to 6 hours. Cool duck with its accumulated juices, wrapped in aluminum foil, to room temperature. Chill in the refrigerator for 12 hours or overnight.

Pick meat from bones and place in a bowl.

Spoon all accumulated juices in the foil into a saucepan and place over medium-high heat. Cook until hot throughout; strain through a fine-mesh strainer into a bowl. Let fat and stock separate; transfer fat from the top to another bowl.

Mash duck meat, 3 tablespoons duck fat, 2 tablespoons duck stock, butter, brandy, parsley, chives, Dijon mustard, cayenne pepper, salt, and ground black pepper together in a bowl with a wooden spoon; transfer to a sealable container, press down, and drizzle a little duck fat over the top. Sprinkle thyme leaves, black pepper, and orange zest over the top. Seal the container and refrigerate for the flavors to blend, 1 to 3 days.