Prepare the Chicken: Generously season the chicken thighs with chopped thyme, smoked paprika, kosher salt, and freshly ground black pepper. Set the Instant Pot to the Sauté function.
Sear the Thighs: Add the olive oil to the heated pot. Place the chicken thighs, skin-side down, into the pot. Cook until the skin turns golden and crispy, about 3 to 5 minutes. If necessary, cook in batches. Flip and sear for an additional 3 minutes. Remove the chicken from the pot and set aside.
Add Aromatics: Introduce the sliced garlic into the pot, stirring until it becomes fragrant, roughly 1 minute. Pour in the chicken broth or water, adding the fresh thyme sprigs. Use a wooden spoon to deglaze the pot, scraping up any browned bits stuck to the bottom.
Pressure Cook the Chicken: Place the trivet into the Instant Pot and arrange the chicken thighs skin-side up on top. Secure the lid and select the manual setting, cooking on high pressure for 12 minutes. Allow the pressure to release naturally.
Make the Gravy: Carefully take out the thighs and trivet. Return the Instant Pot to Sauté mode. Create a slurry by mixing 3 tablespoons of the hot broth with the flour until smooth, then incorporate this mixture back into the pot. Let it simmer until the gravy slightly thickens, around 7 minutes. Discard the thyme sprigs, and adjust seasoning with additional salt and pepper if needed.
Serve: Serve the chicken thighs hot, drizzled with the prepared gravy. Enjoy this comforting dish with your favorite side.