Cook broccoli in boiling water for 3-5 minutes. Drain and rinse with cold water; then set aside. Core tomatoes and cut in half crosswise. Squeeze out seeds. Heat a heavy skillet until very hot. Brush the cut sides of the tomatoes with ½ tablespoon olive oil and place cut side down in the pan. Cook until charred (4-5 minutes). Brush tops of tomatoes with ½ tablespoon olive oil, turn, and cook until charred (1-2 minutes). Transfer tomatoes to a plate to cool. Heat remaining olive oil in skillet; add salt, pepper, and chili powder and cook for 45 seconds. Pour in lemon juice, and then remove from heat. Chop tomatoes and combine with chicken and broccoli. Toss with dressing.