Turn Instant Pot to Saute setting and melt 1 tablespoon butter and 1 tablespoon olive oil in the pot. Add two chicken breasts and cook for 3 minutes per side. Remove chicken from pot to a plate. Cook the remaining chicken and then transfer it to the plate. Set aside.
Add remaining butter and olive oil to the pot. Once butter is melted, add shallots and cook until softened. Add garlic and mushrooms, cook for 1 minute.
Turn off the pot and add in chicken broth, marsala wine, and herbs de Provence. Add chicken breasts back to the Instant Pot as well.
Lock the Instant Pot lid in place and set the valve to sealing. Cook on manual setting (high pressure) for 8 minutes.
When the cooking time is done, quick release the pressure. Remove chicken from the pot and set aside. (I cover it will foil to keep things warm.)
Take out 1/2 cup of cooking liquid and whisk the flour into it.
Turn the Instant Pot to saute. Slowly whisk the flour slurry into the sauce. Keep stirring and cook until sauce has thickened.
Serve chicken warm with mushrooms sauce spooned over the top. Sprinkle some parsley over the top for garnish.