01 - Pat chicken breasts dry with paper towels. Season both sides generously with kosher salt and black pepper. Dredge each breast lightly 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 breasts and sear for 4-5 minutes per side until golden brown and cooked through (internal temperature 165°F). Transfer to a plate and cover loosely with foil.
03 - Add remaining 1 tablespoon butter to the skillet. Add sliced mushrooms and sauté for 4-5 minutes, stirring occasionally, until they release moisture and develop golden brown edges. Add shallot and garlic; cook 1 additional 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. Whisk continuously until cheese melts completely and sauce becomes smooth and velvety. Simmer for 2-3 minutes until sauce thickens enough to coat the back of a spoon.
06 - Return chicken breasts and accumulated juices to the skillet. Spoon sauce and mushrooms generously over the chicken. Simmer gently for 2-3 minutes until everything is heated through. Sprinkle with chopped parsley and serve immediately.