Cut each chicken breast in half horizontally so you have 6 thin chicken breasts. Season with salt and pepper and dredge in flour.
Turn Instant Pot to "Saute" mode. When hot, add the oil. Once the oil heats up, add half the chicken. Cook until lightly browned on both sides. Remove chicken and set aside. Repeat with remaining chicken. Remove and set aside.
Add butter. When melted, add shallots, garlic and mushrooms. Lightly season with salt and pepper. Cook for 3 to 4 minutes.
Return chicken to Instant Pot and add Italian seasoning, red pepper flakes, black pepper, chicken broth, and champagne.
Press "cancel" on Instant Pot. Place lid on. Make sure valve is in sealing position. Set to "Pressure Cook" for 5 minutes. Do a quick pressure release.
Remove chicken. Stir together cornstarch with 2 teaspoons of cold water and add to Instant Pot. Stir in and turn Instant Pot on "Saute". Simmer for 2 to 3 minutes.
Stir in cream and boursin cheese. When smooth, add Parmesan and stir until melted.Ladle sauce over chicken and serve.