I don't have the recipe from Brown Derby, but I have the one from Boma. Usually their recipes are similar from restaurant to restaurant, but not exactly the same. One the WDW site, there is a place to "Contact Us". You might try asking for the recipe there.
Just in case, here's the Boma version:
MULLIGATAWNY
from Boma at Disneys Animal Kingdom Lodge
Yield: 8 servings
6 oz Chicken Stock, add chicken base
2 oz Heavy Cream
1 oz Milk
5 oz Celery, diced
5 oz Onion, diced
8 oz Red Bell Pepper, diced
8 oz Green Bell Pepper, diced
5 oz Carrots, diced
12 oz Apples, frozen and cut in half
1 Tbl Curry Powder, toasted
1 oz Honey
¼ oz Chopped Parsley
1/16 pint Hot Sauce
1 1/16 lb Chicken Breast, diced chunky
Roux (1/8 lb butter and 1/4 lbflour)
Melt butter and stir in flour to make a roux and set aside.
Poach chicken breast in chicken stock until ¾ done, remove, and cut diced, chunky style. Remove excess fat from the stock. Add roux, milk and heavy cream. Let simmer until thicken. Add the celery, carrots, and curry powder. Let cook until vegetables are half done. Add the remaining ingredients including diced chicken.
Let simmer until peppers are tender. Add apples at the last minute. Adjust seasoning. Add parsley when ready to serve.