Rub the turkey breast with 1 tablespoon olive oil, the thyme, 1 teaspoon salt and 1/2 teaspoon pepper. Heat 1 tablespoon olive oil in a large cast-iron skillet over high heat. Add the turkey, skin-side down, and cook until golden, about 3 minutes. Turn and cook 2 more minutes. Reduce the heat to medium high; add the carrots, celery, onion and garlic and cook, stirring, 2 minutes. Add the chicken broth and Worcestershire sauce and bring to a boil. Reduce the heat to medium low, cover and simmer until the turkey is cooked through, about 20 minutes. Transfer the turkey to a cutting board; let rest 5 minutes, then slice.
Preheat the broiler. Drizzle the cut sides of the bread with the remaining 1 tablespoon olive oil. Arrange the bread bottoms on a baking sheet; top each with a slice of cheese. Broil until melted, about 1 minute. Top with the turkey, vegetables and some of the pan juices, then cover with the bread tops. Serve with more pan juices for dipping.