Even the chicken. Pat the chicken dry and, using a meat mallet or rolling pin, flatten each breast to an even thickness (about ¼–½ inch). Cut into portions if you prefer smaller pieces. Season the seasoned flour and lightly coat each piece, shaking off excess.

Brown the chicken. Heat the butter and olive oil in a large skillet over medium-high heat. When foaming, add the chicken and sear until a deep golden crust forms, roughly 4–5 minutes per side. Work in batches if the pan is crowded. Transfer the browned chicken to a plate and set aside.

Cook the mushrooms and garlic. Reduce heat to medium and add a touch more oil if the pan looks dry. Add the sliced mushrooms and cook, stirring occasionally, until they release their liquid and begin to caramelize (about 4–5 minutes). Stir in the garlic for the last 30–60 seconds until fragrant.

Deglaze the pan. Pour in the white wine and scrape up the browned bits from the bottom of the skillet with a wooden spoon. Let the wine bubble and reduce for a minute or two so the flavor concentrates.

Add herbs and simmer. Toss in the thyme sprigs, return the chicken to the skillet, and bring the mixture to a gentle simmer. Cover, lower the heat, and cook until the chicken reaches an internal temperature of 165°F (about 10–15 minutes, depending on thickness).

Finish the sauce. Remove the chicken to a warm plate. Stir the heavy cream into the mushroom pan and heat gently. Add the grated Asiago and stir over low heat until the cheese melts and the sauce thickens to a silky consistency (3–5 minutes). Taste and season with salt and pepper.

Bring it together. Nestle the chicken back into the sauce briefly to heat through. Serve straight from the skillet, finished with extra grated Asiago or a few thyme leaves.