Prepare Chicken:
Season chicken breasts with salt, pepper, and dried thyme.

Cook Chicken:
Heat olive oil in a large skillet over medium-high heat. Add chicken and cook 5–7 minutes per side until golden and cooked through. Remove chicken and set aside.

Cook Mushrooms and Garlic:
In the same skillet, add mushrooms and sauté until browned, about 5 minutes. Add garlic and cook 1 more minute until fragrant.

Make Cream Sauce:
Reduce heat to medium-low. Stir in heavy cream, Asiago cheese, and Parmesan cheese. Cook, stirring frequently, until cheese melts and sauce thickens, about 3–5 minutes.

Combine and Serve:
Return chicken to the skillet, spoon sauce over the top, and cook for another 2 minutes to heat through. Garnish with fresh parsley before serving.