01 - Pat chicken breasts dry with paper towels. Season both sides generously with kosher salt and black pepper. Lightly dredge each piece in flour, shaking off excess coating.
02 - Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat until butter foams. Add chicken and sear for 4 to 5 minutes per side until golden brown and internal temperature reaches 165°F. Transfer to a plate and cover loosely with foil.
03 - Add remaining 1 tablespoon butter to the skillet. Add sliced mushrooms and cook for 4 to 5 minutes, stirring occasionally, until moisture releases and mushrooms begin to brown. Add shallot and garlic; sauté for 1 minute until fragrant.
04 - Pour white wine into the skillet, scraping up browned bits from the bottom with a wooden spoon. Simmer for 2 minutes until slightly reduced. Add chicken broth and bring to a gentle simmer.
05 - Stir in heavy cream and grated Asiago cheese, whisking continuously until cheese melts completely and sauce becomes smooth and velvety. Let simmer for 2 to 3 minutes until sauce thickens slightly.
06 - Return chicken breasts and any accumulated juices to the skillet. Spoon sauce and mushrooms over the chicken. Simmer gently for 2 to 3 minutes until heated through. Garnish with chopped parsley and additional Asiago cheese if desired. Serve immediately.