Position a rack in the middle of the oven and preheat to 500 degrees . Arrange the tomatoes cut side up on a parchment-paper-lined baking sheet and drizzle 2 tablespoons olive oil on top; season with salt and pepper. Roast until tender, about 20 minutes. Transfer the tomatoes and any juices to a bowl.
Meanwhile, in a large saucepan, heat 1 tablespoon olive oil over medium heat. Season the chicken with salt and pepper, add to the pan and cook, turning once, until cooked through, about 10 minutes; transfer to a work surface. Let cool, then shred the meat.
In the same saucepan, heat the remaining 1 tablespoon olive oil over medium heat. Add the onions, season with salt and cook until softened, about 5 minutes. Add the carrots, potatoes and chicken broth and bring to a simmer; cook for 5 minutes. Add the green beans and tomatoes with their juices, partially cover and simmer until the potatoes are tender, about 10 minutes. Stir in the chicken and parsley; season with salt and pepper.