Step 1: In a shallow bowl, whisk together the flour, salt, and black pepper. Dredge each chicken breast in the flour mixture, shaking off any excess.
Step 2: Heat the olive oil and butter in a large skillet over medium-high heat until the butter is melted and foaming.
Step 3: Add the chicken to the skillet and cook for 5-6 minutes per side, or until golden brown and cooked through (internal temperature of 165°F). Remove the chicken from the pan and set aside on a plate.
Step 4: In the same skillet, add the minced garlic and cook for 1 minute until fragrant, being careful not to burn it.
Step 5: Pour in the chicken broth and use a wooden spoon to scrape up the browned bits from the bottom of the pan. Let the broth simmer for 2 minutes to reduce slightly.
Step 6: Reduce the heat to medium-low and stir in the heavy cream and fresh thyme. Allow the sauce to simmer gently for 3-4 minutes until it starts to thicken.
Step 7: Stir in the grated Asiago cheese until melted and smooth. Return the chicken breasts to the skillet, spooning the gravy over them.
Step 8: Garnish with fresh parsley and serve your garlic chicken with asiago gravy immediately over pasta or potatoes.