Prepare Turkey Quarters: Pat dry the turkey quarters with paper towels. Season with generous amount of kosher salt and ground black pepper.
Prepare Pressure Cooker: Heat up your pressure cooker (Instant Pot: press Sauté button and click Adjust button to Sauté More function). Make sure your pot is as hot as it can be (Instant Pot: wait until indicator says HOT).
Brown Turkey Quarters: You can brown them in the Instant Pot, but we found it easier to brown them on a skillet. Add 1 tablespoon (15ml) of olive oil. Place the turkey quarters on a preheated skillet over medium-high heat or in the Instant Pot. Brown each side for roughly 4 – 5 minutes. Remove and set aside.
Deglaze Skillet (if you're using a separate skillet): Deglaze the skillet by pouring ½ cup (125ml) unsalted chicken stock and scrub the flavorful brown bits off the bottom of the skillet with a wooden spoon. Reserve the flavorful liquid.
Sauté Onion and Garlic: Add 1 tablespoon (15ml) of olive oil. Then, add diced onion in the pressure cooker. Sauté for roughly a minute until soften. Add minced garlic and sauté for another 30 seconds until fragrant. Season with kosher salt and ground black pepper.
Sauté Celery and Carrot: Add in chopped celery and carrot. Sauté until the vegetables are slightly browned (roughly 5 – 7 minutes).
Optional layer seasoning: taste and season with more kosher salt and ground black pepper.
Add 2 bay leaves, a pinch of rosemary, sage and thyme (rub the herbs with your fingers to “activate” them), then sauté for another minute until fragrant.
Deglaze: Add a dash of sherry wine in the pressure cooker and deglaze by scrubbing the flavorful brown bits off the bottom of the pot with a wooden spoon.
Add Turkey Quarters & Potatoes: Add browned turkey quarters, reserved flavorful liquid from skillet (if any) and the remaining unsalted chicken stock in the pressure cooker. Place a steamer basket into the pot and add in the halved Yukon gold potatoes. Close lid and pressure cook at High Pressure for 18 – 20 minutes + 10 minutes Natural Release.
After 10 minutes of natural release, turn the Venting Knob to Venting Position to release the remaining pressure. Open the lid carefully.
Make Mashed Potatoes: Remove the steamer basket and place all the potatoes into a large mixing bowl. Mash the potatoes with a fork or masher. Add 1 – 2 tbsp (14g – 28g) unsalted butter, 30g grated Parmesan cheese, and 100ml milk or cream. Continue mashing and mixing until desired consistency. Season with kosher salt and ground black pepper to taste. For creamier mashed potatoes, use more milk/cream.
Make Turkey Gravy: While mashing the potatoes, remove the turkey quarters and set aside. Heat the pressure cooker over medium heat (Instant Pot: Press Saute). Taste the sauce and season with salt and ground black pepper if necessary. Mix 3 tbsp of cornstarch with 2 tbsp of water and stir the mixture into the turkey gravy one third at a time until desired thickness.
Serve: Drizzle the turkey gravy over the turkey quarters and mashed potatoes and serve immediately.