In a small bowl, combine apricots and 1/2 cup cognac. Let stand for 30 minutes. Drain, reserving liquid; set aside.
Heat the oil in a medium skillet over medium heat. Add the onion, shallots, and garlic, and cook, stirring occasionally, until translucent, about 5 minutes. Transfer to a small bowl, and place in freezer until cold, about 15 minutes.
Preheat oven to 325 degrees. In a medium bowl, whisk together the onion mixture, eggs, reserved cognac plus the remaining 1/2 cup salt, pepper, thyme, and allspice; set aside. In a large chilled bowl, combine pork, fatback, ground duck, ground chicken, and apples. Mix well. Add the egg mixture, and mix until well combined.
Line terrines with caul fat leaving a 2-inch overhang. Divide half of the meat mixture between the 4 terrines, gently press to pack down. Arrange 5 apricots down the center of each terrine, with the strips of duck and pistachios on both sides. Top with remaining meat mixture. Fold caul fat over to enclose, and trim if necessary. Cover with parchment paper and then the top of the terrine or aluminum foil.
Place 2 terrines in each of 2 roasting pans. Add enough boiling water to come halfway up the sides of the terrines. Transfer to oven, and bake until a meat thermometer inserted into the center registers 140 degrees. Remove from oven, and uncover.
Place a piece of parchment-paper-lined aluminum foil over each terrine. Cut cardboard to fit just inside the rim of each terrine. Place cardboard on each terrine, and top each with a weight, such as a few cans. Let stand at room temperature for 2 hours. Then refrigerate overnight, and up to 2 days. Unmold or serve directly from the terrine.
Store terrines refrigerated for up to 2 weeks.