1. In a large saucepan, add enough salted water to reach a depth of 2 inches; bring to a boil. Add the asparagus and cook until crisp-tender, about 3 minutes. Drain and rinse with cold water, then pat dry with paper towels.
2. In a nonstick medium skillet, heat 1 tablespoon olive oil over medium heat. Add the garlic and cook, stirring, until golden about 2 minutes. Add the mushrooms, season with salt and pepper and cook, stirring, until tender, about 5 minutes. Cover to keep warm.
3. Preheat a grill or grill pan to medium-high. Cut the beef crosswise into 4 equal portions and pound 1/3 inch thick. Season with salt and pepper, then spread a quarter of the cheese in the center of each portion, leaving a ¾-inch border. Divide the thyme and asparagus equally among each portion. Drizzle each with a teaspoon of olive oil, roll up the meat around the asparagus and secure with toothpicks. Pour the remaining olive oil on a plate and season with salt and pepper. Turn each beef roll in the oil to coat, cover and grill until the cheese is melted; about 10 minutes for medium-rare.
4. To serve, divide the mushrooms among 4 plates; top each with a beef-asparagus roll.