In a oven-proof Dutch oven, combine the cider, fresh garlic (if using), broth, onion, celery, parsley, bay leaf, thyme, salt, paprika and pepper, add in the cubed pork; heat to boiling, reduce heat, simmer partially covered until the pork is tender (about 1 hour).
Mix the evaporated milk with the egg yolk; stir into the pork mixture.
On a small plate, work the butter and flour together; whisk into pork mixture in the pot.
Cook, stirring constantly until thickened.
Stir in lime juice.
Heat the oven to 350 degrees.
Transfer the pork mixture into a 1-1/2 quart baking dish.
Arrange the sliced apples evenly over the top.
Sprinkle with cheese.
Bake for 20-35 minutes, or until the cheese is melted and lightly browned.