Rinse chicken livers under cold running water; remove fat and connective tissue. In a skillet heat oil. Cook livers with mushrooms, apple, onion, garlic, thyme, salt, and pepper until livers are browned and cooked through, stirring often. Add cider or juice; reduce heat. Simmer, uncovered, until liquid is nearly evaporated. Add milk. Cook, uncovered, for 3 minutes or until slightly thickened.
Transfer to a food processor bowl. Cover; process until smooth. Press mixture through strainer. Spoon into serving dish or individual ramekins. Cover and chill for at least 6 hours or for up to 3 days. Makes about 1-3/4 cups (6 servings).