Turn the Instant Pot to Saute, and add the olive oil to the pot. Season chicken with salt and pepper and sear in the Instant Pot until golden brown. Remove and set on a plate.: Warmth hits the pan first, and you will smell the oil bloom then the satisfying sizzle as the chicken skin kisses the hot surface, releasing a rich, nutty aroma and creating browned bits, or fond, at the bottom of the pot. This fond is pure flavor, it will dissolve into the sauce later, enriching it with savory depth. As you sear aim for a deep golden brown rather than black, flipping once so each side develops color. If the pot seems crowded steam will form instead of a crisp sear, so work in batches if necessary. A common mistake is rushing this step, which yields pale meat and a thinner sauce, so give it time until the skin looks caramelized and slightly crisp. After searing, the chicken should feel firmer and emit a roasted scent; set it aside to preserve that crust while you build the sauce.
Add onion and smashed garlic cloves to the Pot and saute until soft and translucent. Add tomato paste and stir for two minutes. Add chopped tomatoes, chicken broth, thyme, olives, and capers.: As the onion sweats you will notice a sweet, savory perfume rise and the pieces go from raw and sharp to soft and glossy, which mellows bitterness and develops sweetness that balances the acidic tomatoes . The smashed garlic will release an aromatic oil that coats the pan, and when you add the tomato paste it should darken slightly and smell richer, a sign that its sugars are caramelizing, adding umami. Pour in the chopped tomatoes and the low sodium chicken broth, and you will see the liquid loosen the paste into a cohesive base. Sprinkle in the fresh thyme , scatter the Kalamata and green olives , and stir in the capers so their brininess disperses. At this point taste a little of the broth, but avoid heavy salting because the olives and capers bring saltiness. If you skip properly softening the onion the sauce can taste flat, so ensure the onion is translucent and fragrant before moving on.
Return the chicken to the Instant Pot, lock the lid, and set the Manual or Pressure Cook setting for 25 minutes.: Once the chicken nestles into the tomato mixture you will notice the aroma deepen as the herbs and brine mingle with the meat. Locking the lid traps steam, which forces flavors into the bones and fibers quickly, producing tender meat with concentrated sauce. The pressure cook time is calibrated so the thighs and drumsticks become fork tender without disintegrating. A tip is to ensure the pieces are in a single layer where possible so heat distributes evenly; overcrowding can cause uneven cooking. If you do not release pressure correctly you risk overcooking, so plan for a controlled release following the cook time.
Release pressure manually, then remove the lid with care. Remove the chicken and set aside. Turn on the saute setting and simmer the sauce an additional 10 minutes, until the sauce reduced to about a pint. Add chopped parsley and lemon juice, then pour over chicken.: When you release the pressure the aroma will bloom suddenly, sharp and savory, and steam will escape, carrying concentrated scents of herb and tomato. Remove the chicken gently onto a plate so it can rest while you reduce the sauce, which intensifies flavor and thickens texture. As the sauce simmers it will bubble steadily, and the surface will darken and glossy droplets will cling to the spoon, indicating the right consistency. Stir occasionally to deglaze the pan and dissolve any fond, which adds richness. Before serving fold in fresh parsley and a splash of lemon juice to brighten the sauce, balancing the salty elements. A common misstep is rushing the reduction, leaving the sauce watery; simmer until it coats the back of a spoon and tastes concentrated.
Serve over rice, pasta, or polenta.: The final presentation lets the sauce sing, and you will notice steam rising as the glossy tomato mixture pools around the chicken . The starch you choose will absorb the sauce differently, rice soaks it up, pasta twirls it, and polenta offers a creamy bed. Spoon sauce generously so each bite marries meat and tomato, and garnish if desired with extra parsley for color. Avoid plating with chilled starch that will stiffen the sauce; always serve warm to maintain the right texture and aroma.