Preheat the oven to 375°F (190°C). Position an oven rack in the center.
Pat chicken breasts dry with paper towels. Season both sides with salt and pepper evenly.
Spread the grated Asiago cheese on a plate and press each chicken breast into the cheese to coat both sides, ensuring a thin, even layer sticks.
Heat 1 tablespoon olive oil in a large oven-safe skillet over medium-high heat until shimmering.
Add the Asiago-coated chicken to the skillet. Sear without moving for 4–5 minutes until the cheese crust is golden brown and releases easily from the pan.
Flip the chicken and sear the other side for another 4–5 minutes to form a crust.
Transfer the skillet to the preheated oven and bake for 15–20 minutes, or until the internal temperature reaches 165°F (74°C) and juices run clear.
Carefully remove the skillet from the oven and transfer the cooked chicken to a plate; tent with foil to keep warm.
Place the skillet back on the stove over medium heat. Add the minced garlic and sauté for about 1 minute until fragrant; do not brown.
Pour in 1/2 cup heavy cream and add 2 tablespoons Dijon mustard. Stir to combine, scraping up any browned bits from the bottom of the pan.
Simmer the sauce for 3–4 minutes, stirring occasionally, until it thickens slightly and becomes silky.
Taste and adjust seasoning with salt and pepper as needed. Return the chicken to the skillet and spoon sauce over each piece.
Garnish with 1 tablespoon chopped fresh parsley and serve immediately.
Notes
For a lighter sauce, swap half-and-half for heavy cream, but expect a slightly thinner sauce.