Pat the chicken breasts dry and season both sides generously with salt and pepper.
Heat 1 tablespoon of olive oil and 1 tablespoon of butter in a large skillet over medium-high heat. Add the chicken and cook for 5-6 minutes per side until golden brown and cooked through (internal temperature 165°F). Remove chicken to a plate and set aside.
In the same skillet, add the remaining butter and mushrooms. Sauté for 5-7 minutes until mushrooms are browned and moisture has evaporated.
Add the minced garlic and thyme, cooking for 1 minute until fragrant.
Pour in the white wine to deglaze the pan, scraping up any browned bits from the bottom with a wooden spoon. Let it simmer for 2-3 minutes until reduced by half.
Stir in the chicken broth and heavy cream. Bring to a gentle simmer (do not boil vigorously) and cook for 3-4 minutes until the sauce slightly thickens.
Reduce heat to low. Stir in the grated Asiago cheese until completely melted and smooth.
Return the chicken and any accumulated juices to the skillet. Spoon the sauce over the chicken to reheat for 1 minute.
Garnish with fresh parsley and serve immediately.