Prepare the Brine: In a large container, dissolve 1 ½ tablespoons salt in 2 quarts cold water. Add beans and soak at room temperature for 8 to 24 hours. Drain and rinse well.
Sauté Aromatics: Using the highest sauté or browning function on the Instant Pot, heat the olive oil until just smoking. Add pancetta and cook, stirring often, until crisp (about 5 minutes). Stir in onion, celery, and carrots, and cook until softened, about 5 minutes. Add garlic and cook until fragrant, about 30 seconds.
Combine Ingredients: Stir in chicken broth, scraping up any browned bits. Add the beans, kale, diced tomatoes (with juice), bay leaves, and rosemary sprig.
Cook in Instant Pot: Lock the lid in place and close the pressure release valve. Select the high-pressure cook function and cook for 3 minutes.
Natural Pressure Release: Turn off the Instant Pot and let the pressure release naturally for 15 minutes. Quick-release any remaining pressure, then carefully remove the lid, allowing steam to escape away from you.
Finish and Serve: Discard the bay leaves and rosemary sprig. Adjust the stew's consistency with extra hot broth, as needed. Season with salt and pepper to taste. Drizzle individual servings with olive oil before serving.