Pound chicken breast with a meat mallet between 2 sheets of waxed paper or plastic wrap until meat is uniform thickness; about ¼-inch. Cut into serving-sized pieces (2 or 3 pieces per breast).
Heat the butter with 1 tbs of olive oil in deep, heavy skillet or saute pan over medium heat.
Dredge chicken in flour, that has been mixed with the salt and pepper. Add to hot oil and butter mixture in skillet. Saute until golden on each side, about 5 minutes per side. Remove from pan.
Add remaining olive oil to hot skillet. Saute mushrooms and garlic until mushrooms begin to brown.
Add white wine to skillet, scraping up all the browned bits that are at the bottom of the pan. Bruise the fresh thyme by twisting it between your fingers or hitting it with the dull side of a knife in a few places. Add thyme to mushroom and wine mixture in pan.
Add the chicken back to the pan. Bring to a boil, then reduce heat, cover and simmer for 15-20 minutes.
Remove chicken from pan. Add the cream and heat through. Add the asiago cheese- add more cheese for a stronger flavour.
Cook, stirring constantly over low heat until cheese melts. Continue cooking until sauce is reduced by about half.
Add chicken back to pan and heat through.
Garnish with sprigs of fresh thyme. May be served over any pasta.