Grab your chicken thighs and sprinkle ¾ teaspoons kosher salt all over them. You can do this part up to two days in advance, and keep the chicken covered in the fridge.
Press the "Sauté" button on the Instant Pot or heat a stovetop pressure cooker over medium heat. Add your fat of choice.
When the fat is shimmering, add the onions, carrots, and mushrooms, and a ½ teaspoon kosher salt. Sauté the vegetables until the softened, about 3 to 5 minutes.
Stir in the garlic and tomato paste, and cook for about 30 seconds or until fragrant.
Add the salted chicken, cherry tomatoes and green olives to the pot. Give it all a good stir.
Turn off the sauté function, and lock the lid on the pressure cooker.
If you’re using an Instant Pot, press the “Manual” or “Pressure Cook” button and set the cooking time at 10 minutes for boneless thighs. Other chicken cuts? Program 7 minutes under high pressure for boneless chicken breasts (if your chicken boobs are bigger than 8 ounces each, you can cut them in half or increase the cooking time to 9 minutes) and 15 minutes under high pressure for bone-in thighs and drumsticks.(For those of you using a stovetop pressure cooker, crank the heat to high to bring the contents to high pressure. Once high pressure is reached, reduce the heat to low, or just enough to maintain high pressure. Set a timer for 6 minutes for breasts and 9 minutes for thighs.)
When the chicken is finished cooking, release the pressure manually.
Remove the lid right away to prevent overcooking. Add the freshly cracked black pepper and adjust the seasoning with additional salt as needed.
If you want a thicker gravy, remove the chicken and blend up the vegetables and cooking liquid with an immersion blender.
Stir in the fresh herbs, and serve immediately.
Toss any leftovers (covered, of course) in the fridge for up to 4 days or in the freezer for up to 6 months.