When you're not feeling quite up to par, this broth, subtly flavored with slivers of shiitake mushrooms and tender broccoli florets, is a nourishing and tasty meal.
Combine chicken wings, chicken stock, dried shiitake mushrooms, onion, thyme, and peppercorns in a medium saucepan. Cover; bring liquid to a boil. Reduce heat; simmer gently, with lid slightly ajar, 1 hour.
Strain the broth through a fine sieve set over a heat-proof bowl, and discard the solids. Let the broth cool to room temperature, and cover bowl with plastic wrap. Transfer broth to refrigerator, and chill.
Heat oven to 400 degrees. Wrap garlic head in aluminum foil, and cook until cloves become very soft, 35 to 45 minutes. Set roasted garlic aside to cool.
Using a spoon, skim off all fat from chilled broth; discard the fat. Transfer the broth to a medium saucepan, and warm over medium-low heat. Add the fresh shiitake mushrooms; continue warming soup.
Remove the roasted garlic from each clove, and transfer to a small bowl. Using a fork, combine until a puree forms. Set aside.
Place a steamer basket in the bottom of a large saucepan containing 1 inch of water. Place broccoli florets in steamer. Cover, and bring to a boil over high heat. Reduce heat to medium low, and steam, until broccoli is fork tender, 1 to 3 minutes. Remove broccoli from the steamer.
Add steamed broccoli to the soup. Season with salt and pepper. Divide soup among four soup plates, and garnish each with 1/2 teaspoon roasted-garlic puree. Serve.
You can make the broth up to Step 2 a couple of days in advance, and refrigerate.