Pat the chicken breasts dry with paper towels. Sprinkle them all over with the salt and pepper.
Select the Sauté setting on the Instant Pot and heat the oil for 2 minutes.
Using tongs, add the chicken in a single layer and sear for about 2 minutes, until lightly browned. Flip the chicken and sear for 2 more minutes, until lightly browned on the second side, then transfer to a plate.
Add the garlic, onion, mushrooms, and sauté for about 4 minutes, until the onion begins to soften and the mushrooms begin to wilt and give up their liquid.
Add the Italian seasoning, wine, and tomatoes and stir to combine, using a wooden spoon to deglaze any browned bits from the base of the pot. Let the wine come to a simmer, then add the chicken breasts in a single layer on top, then press cancel.
Secure the lid and select Pressure Cook and set the cooking time for 5 minutes
When the cooking program finishes, allow the pressure to release naturally for 5minutes, then quick release the remaining pressure, and press cancel.
Open the pot and, using tongs, transfer the chicken to a serving plate.
Select the Sauté setting again, then add the spinach and basil and stir until wilted, about 1 minute.
Add the cream cheese and Pecorino Romano cheeses and stir for 1 minute, until melted.
In a small bowl, stir together the cornflour and water, then add to the pot, and stir for another minute until the sauce is slightly thickened.
Spoon the sauce and vegetables over the chicken and serve right away.