01 - Pat the chicken breasts dry with paper towels, then season both sides generously with kosher salt and freshly ground black pepper. Dredge each breast lightly in flour, shaking off any excess coating.
02 - Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat until butter foams. Add chicken breasts and sear for 4 to 5 minutes per side until golden brown and cooked through. Transfer to a plate and cover loosely with foil.
03 - Add remaining 1 tablespoon butter to the skillet. Add sliced mushrooms and sauté for 4 to 5 minutes, stirring occasionally, until they release moisture and develop golden-brown edges. Stir in shallot and garlic; cook for 1 minute until fragrant.
04 - Pour white wine into the skillet, scraping up any browned bits from the bottom with a wooden spoon. Let simmer for 2 minutes until slightly reduced, then add chicken broth and bring to a gentle simmer.
05 - Stir in heavy cream and grated Asiago cheese, whisking continuously until the cheese melts completely and the sauce becomes smooth and velvety. Let simmer for 2 to 3 minutes until slightly thickened.
06 - Return chicken breasts and any accumulated juices to the skillet. Spoon sauce and mushrooms generously over the chicken. Simmer gently for 2 to 3 minutes until heated through. Sprinkle with chopped parsley and serve immediately.