Cut each chicken breast into cutlets. Lightly season each cutlet on both sides with salt and pepper and dredge in flour, shaking off any excess.
Place a large skillet over medium-high heat and melt about 2 Tbsp unsalted butter. Add as many cutlets as will fit comfortably into the skillet, but be sure not to overcrowd the pan. Sauté for about 5 minutes until golden brown on the first side, then flip and cook for 1 additional minute. Remove the chicken from the skillet to a plate and cover lightly with foil. Repeat with remaining chicken cutlets.
When all of the chicken is cooked and set aside, add remaining butter to the skillet along with the mushrooms and ½ tsp salt. Cook for about 10 minutes, stirring occasionally, until the mushrooms are golden brown and become softened. Add garlic and cook for an additional 2 minutes.
Pour white wine (or ½ cup chicken broth) into the skillet and stir to scrape up anything that is sticking to the bottom of the pan. Once the wine has nearly all evaporated, add ¾ cup chicken broth and thyme. Bring the mixture to a boil, reduce heat to medium and cook for 10–15 minutes at a rapid simmer until the liquid is reduced by half.
Reduce heat to medium-low and stir in heavy cream. Sprinkle the Asiago cheese over the top of the sauce and stir constantly until the cheese melts. Nestle the chicken back into the skillet and allow it to simmer in the sauce just until the chicken is heated through and the sauce has slightly thickened. Sprinkle with a bit of Parmesan cheese and serve right away, either all by itself, or spooned over pasta or mashed potatoes. Enjoy!
Be patient with step 4 in the recipe! You'll want to let the sauce cook down until it is reduced by half before adding the chicken back in. This way it is more flavorful and the chicken won't be overwhelmed by too much sauce.