Add 1/2 cup broth and thyme. Cover, reduce heat, and simmer 5 minutes.
Combine remaining 1/2 cup broth and flour, stirring well.
Add to vegetable mixture, stirring well. Cook, stirring constantly, until slightly thickened and bubbly.
Prepare grill.
Rub steaks with garlic halves, and sprinkle with 1/2 teaspoon black pepper. Coat grill rack with cooking spray; place on grill over medium-hot coals (350 to 400).
Place steak on rack, and grill, covered, 4 minutes on each side or to desired degree of doneness. Spoon vegetables evenly onto individual serving plates; arrange steaks over vegetables.