Set a 6-quart Instant Pot to high saute mode and allow it to preheat for 5 minutes.
Season the chicken generously on both sides with kosher salt and freshly ground black pepper.
Heat olive oil in the Instant Pot. Add half the chicken skin-side down and cook until deeply golden, about 6-8 minutes. Flip and cook for another 4-6 minutes. Remove to a plate and repeat with the remaining chicken.
Add the garlic to the pot and cook, stirring, for 1 minute until fragrant and lightly browned.
Stir in the kalamata olives, capers, white vinegar, oregano, and 1/2 cup of water. Use a wooden spoon to scrape up the flavorful browned bits at the bottom of the pot.
Turn off the heat. Place the Instant Pot rack over the olive mixture, ensuring the handles face up.
Nestle the chicken thighs, skin-side up, on the rack. Secure the lid and set the Instant Pot to high pressure for 10 minutes.
While the chicken cooks, whisk together the Greek yogurt, parsley, mint, lemon zest, lemon juice, and a pinch of salt and pepper in a medium bowl. Adjust seasoning as needed and set aside.
Once the pressure-cook cycle is complete, perform a quick release following the manufacturer’s instructions. Carefully remove the lid.
Remove the chicken and set it on a serving platter. Use a slotted spoon to transfer the olive and caper mixture over and around the chicken. Cover with foil to keep warm.
Remove the rack and set the Instant Pot to normal saute mode. Simmer the sauce for about 6-8 minutes until it reduces by half.
Spoon the reduced sauce over the chicken. Serve with the yogurt sauce on the side for dipping.