Season the chicken with salt and pepper. Heat a large skillet over medium-high heat; add 2 tablespoons olive oil. Add the chicken skin-side down and cook until browned and crisp, about 8 minutes. Flip and continue cooking until browned on the other side, about 3 more minutes. Remove to a plate and pour out the drippings.
Add the remaining 1 tablespoon olive oil, the garlic and red pepper flakes to the skillet. Cook over medium-high heat until the garlic is golden, about 1 minute. Add the wine and scrape up any browned bits from the bottom of the pan. Add the tomatoes, chicken broth and ½ teaspoon salt and bring to a simmer. Add the chicken skin-side up and reduce the heat to medium low. Cover and gently simmer until the chicken is cooked through and the sauce is slightly thickened, about 15 minutes.
Meanwhile, preheat the broiler. Bring a medium saucepan of salted water to a boil. Add the broccolini and cook until crisp-tender, about 3 minutes; drain. Put the ciabatta on a baking sheet and top each slice with a piece of cheese. Broil until the cheese melts, 2 to 3 minutes. Divide the chicken among plates; spoon the sauce over the top. Serve with the broccolini and cheese toasts.