Cream of chicken with wild rice soup

Soup is my cold-weather day food.  Cream of chicken has always been a favorite, and when wild-rice is added in, it gives it texture and a nutty flavor that can't be beat.  Finishing the soup off with white wine gives it a depth that, in my opinion, makes this soup go from 4 to 5 stars.  I buy those mini 4-packs of white wine to have around when cooking instead of having to open up a big bottle.  This is modified off an old recipe I found somewhere years ago and it is an absolute winner.

1 1/3 c wild rice, uncooked
meat from a 3 lb roasted chicken
~1c chopped celery
~1c chopped onion
~1 c chopped carrot (I use the frozen ones!)
~1 c sliced mushroom
2T oil
6c chicken broth
1/2 c butter or margarine
3/4+ c flour
4 c skim milk
1c white wine
+salt (or chicken bouillon) and pepper as needed

1. Cook rice as directed on package in either water or chicken broth (or water + bouillon).  Stop ~10 minutes before end of cooking time.
2. In the meantime, clean chicken off bone, removing fat/gristle and cutting into bite-sized pieces.
3.  Saute celery, onion, and carrot in a very large pot in ~2T oil on medium heat.  After ~5 minutes, add mushrooms and continue to saute ~10 more minutes.
4.  Add chicken broth and set to simmer.
5.  If any excess liquid remains in the rice, drain it.  Rice should be beginning to crack open.  Add drained rice to soup.
6.  In a separate pan, make a roux - melt the butter over medium-high and then stir in the flour to make a paste.  Slowly add milk, one bit at a time at first, stirring thoroughly until it's smooth.  Repeat.  This is a slow process.  If you get lumps, use a whisk to smooth it out, but also use a spoon to get all the flour stuck to the bottom of the pan.
7.  Add some of the hot soup broth to the bechamel to thin it, then whisk it all into the soup.
8.  Finish soup with chicken, white wine and adjust flavorings by adding salt (or chicken bouillon) and pepper.  Allow to simmer for ~10-15 more minutes.