Pat dry the turkey quarters with paper towels. Season with generous amount of kosher salt and ground black pepper.
Press Sauté button and click Adjust button to Sauté More function. Wait until indicator says “hot”.
Brown turkey in the Instant Pot or on a skillet. Add 1 tablespoon 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 the skillet or Instant Pot by pouring 1/2 unsalted chicken stock and scrub the flavorful brown bits off the bottom of the skillet with a wooden spoon. Reserve the flavorful liquid.
Add 1 tablespoon of olive oil to the pot. Add diced onion. 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.
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.
Add a dash of sherry wine in the pot and deglaze by scrubbing the flavorful brown bits off the bottom of the pot with a wooden spoon.
Add browned turkey quarters, reserved flavorful liquid (if any) and the remaining unsalted chicken stock in the Instant Pot. Place a steamer basket into the pot and add in the halved Yukon gold potatoes.
Close lid and pressure cook on Manual at High Pressure for 18 to 20 minutes. Allow for 10 minutes Natural Release when time is up. After 10 minutes of natural release, turn the Venting Knob to Venting Position to release the remaining pressure. Open the lid carefully.
Remove the steamer basket and place all the potatoes into a large mixing bowl. Mash the potatoes with a fork or masher.
Add butter, Parmesan cheese, and 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.
While mashing the potatoes, remove the turkey quarters and set aside. Heat the Instant Pot by pressing Saute. Taste the sauce and season with salt and ground black pepper if necessary.
Mix 3 tablespoons of cornstarch with 2 tablespoons of water and stir the mixture into the turkey gravy one third at a time until desired thickness.
Drizzle the turkey gravy over the turkey quarters and mashed potatoes and serve immediately. Enjoy!