01 - Pat chicken breasts dry with paper towels. Season both sides generously with salt and pepper. Dredge each piece 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 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 mushrooms and cook for 4 to 5 minutes, stirring occasionally, until they release moisture and start to brown. Add shallot and garlic; cook for 1 minute until fragrant.
04 - Pour in white wine, scraping up browned bits from the bottom with a wooden spoon. Simmer for 2 minutes until slightly reduced, then add chicken broth and bring to a gentle simmer.
05 - Stir in heavy cream and Asiago cheese, whisking continuously until cheese melts completely and sauce becomes smooth. 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 everything is heated through.
07 - Sprinkle with chopped parsley. Serve immediately, offering additional Asiago cheese and parsley at the table for garnish.