In a large skillet cook mushrooms and garlic in hot olive oil and butter over medium heat until mushrooms are tender and liquid has evaporated. Stir in thyme, 1/4 teaspoon salt, and 1/8 teaspoon pepper. Remove from skillet; keep warm.
Lightly season tofu slices with additional salt and pepper. In the same skillet cook tofu slices, half at a time, if necessary, over medium-high heat for 3 to 4 minutes or until lightly browned and heated through, turning once. Remove from skillet. Add balsamic vinegar to skillet; bring to boiling. Boil gently for 1 minute. Remove from heat.
To serve, arrange greens on serving plates. Top each with tofu slices. Drizzle with balsamic vinegar. Top tofu with mushrooms and cheese. Serve with bread slices.