Set the Instant Pot to Sauté. Season the chicken thighs generously with kosher salt and black pepper. Heat the olive oil and butter, then sear the chicken for 4–5 minutes per side until golden brown. Transfer to a plate.
Add the onion and cook for 2–3 minutes. Stir in the garlic and cook for 30 seconds. Pour in 2 cups of the chicken broth, scraping up any browned bits from the bottom of the pot.
Stir in the Dijon mustard, whole grain mustard, thyme, and oregano until combined. Return the chicken to the pot, skin-side up.
Lock the lid and cook on High Pressure for 10 minutes. Quick release the pressure, then transfer the chicken to a clean plate.
Turn the Instant Pot back to Sauté. Add the orzo, mushrooms, and remaining 1 cup of chicken broth. Cook, stirring frequently, until the orzo is al dente.
Turn off the heat. Stir in the spinach, heavy cream, Parmesan, lemon zest, and lemon juice until the spinach has wilted and the sauce is creamy. Season with additional salt and pepper to taste.
Spoon the creamy orzo into bowls and top with the chicken thighs. Garnish with extra Parmesan, fresh parsley, or cracked black pepper, if desired.
Notes
Optional step: Once the chicken thighs are removed from the instant pot after their 10 minute cook, place them under the broiler for a few minutes to crisp up the skin.