Generously salt and pepper chicken breast, both sides. In a large, heavy skillet, at medium-high/high heat add butter and chicken breast, and let brown very well for a few minutes on each side. Remove from pan and set aside on a plate. (They are not done at this point, just browned on the outside.)
Add garlic to skillet, saute until brown.
Once garlic is browned, add white wine and stock, simmer and reduce by half.
Once reduced, add mushrooms, milk, heavy cream, cornstarch slurry, both cheeses, and salt and pepper. Mix to combine. Add chicken breast back to pan, reduce heat, and simmer until chicken is cooked through, about 20 minutes (depending on thickness).
Serve over rice.