Slice the chicken breasts horizontally. Place between sheets of wax paper and pound with a meat mallet until very thin (about 1/4"). Cut into large strips.
Add the butter and 1 tablespoon of the olive oil to a large skillet over medium heat.
Place the flour in a pie plate and season with 1 teaspoon of salt and 1/2 teaspoon of pepper. Stir to combine.
Dredge the chicken pieces in the flour then transfer to the skillet. Saute until golden brown on all sides. Remove to a plate.
Add the remaining olive oil to the skillet.
Add the mushrooms and saute until just starting to soften.
Add the garlic and saute for another minute
Add the wine and scrape up any bits on the bottom of the pan.
Bruise the thyme (if using fresh, otherwise, crumble the dried thyme) and add to the pan.
Return chicken to the pan. Bring to a boil then reduce to a simmer. Cover and let simmer for 15 minutes.
Remove chicken pieces to a plate again.
Stir in the cream and cheese until melted. Reduce by half, stirring often.
Add the chicken back and heat thoroughly.
Serve over warm egg noodles, pasta, or rice.