Heat 2 tsp olive oil in a Dutch oven over medium heat. Add fennel and onion; cook, stirring frequently, until tender, about 15 minutes. Add garlic and cook 15 seconds, stirring constantly. Remove from heat.
Combine fennel mixture, 1 C. chicken broth and pumpkin in a blender. Puree until smooth, scraping down sides occasionally with a spatula.
Return fennel mixture to pan and add remaining 2 C. broth and thyme. Cook over medium-low heat 10 minutes, stirring frequently. Season with salt.
Heat a skillet over medium heat. Add remaining 1 tsp olive oil and tilt skillet to coat bottom. Add mushroom caps and cook 5 minutes, stirring frequently.
To serve, spoon soup into individual bowls and sprinkle with mushrooms. Using a vegetable peeler, shave cheese over each serving.
*If you don't like fennel, substitute 2 leeks (white part only) or 3 ribs of celery.