Mix flour with 1 tsp salt and 1/2 tsp pepper in a shallow bowl.
Dredge chicken breasts in the seasoned flour until evenly coated.
Heat olive oil and 1 tbsp butter in a large skillet over medium-high heat.
Sear chicken for 5-6 minutes per side until golden brown and cooked through; remove from pan and set aside.
In the same skillet, add remaining butter and fresh mushrooms; sauté until browned.
Add minced garlic and fresh thyme sprigs; cook for 1 minute until fragrant.
Deglaze the skillet with apple cider vinegar, scraping up any browned bits, and let it reduce by half.
Lower heat to medium-low, stir in heavy cream and asiago cheese until smooth and melted.
Return the chicken to the skillet, coating it in the sauce, and simmer for 2 minutes before serving.