Prepare the chicken first. Place the chicken breasts between two pieces of plastic wrap or parchment paper and use a meat mallet or rolling pin to pound them to an even thickness of about half an inch. This step is crucial for even cooking. Season both sides generously with salt and pepper.

Heat the oil or butter in a large skillet over medium-high heat. Once the pan is hot and the butter is foaming, carefully lay the chicken breasts in the skillet. Do not overcrowd the pan. Cook for 5 to 7 minutes per side, or until the chicken is deeply golden brown and cooked through. The internal temperature should reach 165 degrees Fahrenheit. Remove the chicken to a plate and set it aside.

In the same skillet, lower the heat to medium. Add the sliced mushrooms. Let them cook undisturbed for a few minutes to get some color. Then, stir them and add the minced garlic. Cook for another 2 to 3 minutes, until the mushrooms are browned, tender, and fragrant.

Pour the heavy cream into the skillet, scraping up any browned bits from the bottom of the pan. Those bits are pure flavor. Stir in the Dijon mustard and then add the grated Asiago cheese. Reduce the heat to low and let the sauce simmer gently for 3 to 5 minutes, stirring occasionally, until it thickens slightly and coats the back of a spoon.

Return the chicken and any accumulated juices to the skillet. Spoon the creamy sauce over the top of each piece of chicken. Allow everything to warm through together for an additional 2 to 3 minutes.

Serve hot, spooning extra sauce over the chicken.