A recipe inspired by Brat cheese sauce/Cooking in Wine
I wouldn't use mozzarella, it won't melt smoothly into a sauce. Look at recipes for white (or whitish) smooth cheese sauces, or Mornay sauces. Consider:


Monterey Jack 
Swiss (Emmental)
White Cheddar
Gruyere (that would be my choice)


Instead of boiling chicken (or sausage) in the wine, consider browning the chicken in the skillet (preferably not non-stick) and then deglazing with the wine. After all of the brown goodness is mixed into the small amount of wine used for deglazing, you can add more wine (as much as you want to use for the final sauce) to allow it cook out a bit. You might want to use 1/2 or 1/3 wine, the rest chicken broth and possibly cream.

Look at white wine cheese sauce recipes to get an idea of quantities. To get a thick, smooth sauce, you might get your best results by thickening your liquid with a roux before adding shredded cheese, much like the recipes above for Mornay sauce.