Heat 1 tablespoon oil in a Dutch oven over medium-high heat. Combine squash and next 5 ingredients (squash through garlic), and sauté for 8 minutes or until lightly browned, stirring frequently. Add bay leaves and 2 thyme sprigs, and stir in tomato paste. Stir in flour and wine, reduce heat to medium-low, and cook for 5 minutes. Stir in the broth and chickpeas. Cover and simmer 20 minutes or until vegetables are tender. Stir in 1/2 teaspoon salt, pepper, and chopped parsley. Discard bay leaves and thyme.
Discard mushroom stems; slice caps. Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Add sliced mushrooms, and cook 5 minutes, stirring constantly. Sprinkle with 1/4 teaspoon salt. Serve ragout over mushrooms. Garnish with additional thyme sprigs, if desired.