Dredge (coat) the chicken cutlets in the flour mixture on both sides so they’re fully and lightly dusted.
Add the oil to an Instant Pot, hit Saute, and Adjust to the More or High setting. After 3 minutes of the oil heating, add the chicken in batches (2 to 3 at a time will fit) and lightly brown for 1 minute on each side. Transfer to a plate and repeat until all the chicken is lightly browned.
Add the butter to the pot and scrape the bottom of any flour remnants. Add the shallots and garlic and saute for 2 minutes. Add the mushrooms (if using) and saute for another 2 minutes. Add the Champagne (or prosecco) and deglaze, then add the broth, Italian seasoning, and oregano and stir well. Return the chicken to the pot. Top with the cherry tomatoes and the spinach, but do not stir! Just let it all rest on top.
Secure the lid and move the valve to the sealing position. Hit Cancel and then hit Manual or Pressure Cook on High Pressure for 5 minutes. Quick release when done.
Meanwhile, make a slurry by mixing together the cornstarch and cold water in a small bowl until smooth.
Using tongs, transfer the chicken to a serving dish. Hit Cancel and then hit Saute and Adjust to the More or High setting to bring the pot to a bubble. While stirring, add the cornstarch slurry to the sauce. Let it bubble for 30 seconds while stirring constantly, then hit Cancel to turn the pot off.
Add the Boursin (or cream cheese) and stir until well combined. Add the cream and stir in the grated Parmesan and seasoned salt. Ladle the sauce over the chicken in the serving dish and serve.