Preheat the oven to 375 degrees F, ensuring one rack is positioned in the center of the oven. Gather & prep all ingredients according to ingredients list.
Heat the olive oil in a heavy-bottomed oven-safe skillet over medium-high heat. Pat the chicken thighs as dry as possible, then season well with S&P. Once oil is hot & simmering, brown the chicken. Work in batches as needed. Place the chicken thighs skin side down. Cook 5-6 minutes, until golden brown & rendered. Flip the chicken & cook 3-4 minutes more, until golden brown. Transfer to a plate & set aside.
Add the shallots to the skillet. Season with a good pinch of kosher salt & ground black pepper as desired. Stir to coat in the chicken drippings. Deglaze the skillet with lemon juice & chicken broth, using a wooden spoon to scrape up any browned bits from the bottom of the pan. Nestle the chicken thighs back into the skillet, along with the crushed olives.
Transfer the olive chicken skillet to the oven. Roast 15-20 minutes, or until an instant-read thermometer inserted in the thickest part of the thigh without hitting bone registers an an internal temperature of 165 degrees F.