Rinse chicken livers under cold running water; remove fat and connective tissue. In a skillet heat olive oil. Cook livers with mushrooms, apple, onion, garlic, thyme, salt, and pepper in oil until livers are browned with only slightly pink centers, 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 liver mixture to a food processor bowl. Cover and process until smooth. Press mixture through a strainer. Spoon into a 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.