Notes: Garnish soup with parsley.
1. In a 1- to 2-quart pan, bring 1 1/4 cups broth and rice to a boil over high heat. Reduce heat, cover, and simmer until rice is just tender to bite, 45 to 50 minutes.
2. Meanwhile, in a bowl, soak shiitake mushrooms in 1 1/2 cups hot water until pliable, about 20 minutes. Squeeze mushrooms in water to loosen grit. Lift out mushrooms, reserving liquid. Cut off and discard stems, then thinly slice caps.
3. In a 5- to 6-quart pan over medium-high heat, stir shiitakes in 1 tablespoon butter until golden brown, about 5 minutes. With a slotted spoon, transfer shiitakes to a small bowl.
4. To large pan, add remaining 1 tablespoon butter, onion, common mushrooms, garlic, rosemary, and thyme; stir often until vegetables are golden, about 10 minutes.
5. Carefully pour shiitake soaking liquid into pan, discarding grit in bottom of bowl. To pan, add potato and remaining 2 cups broth. Cover, bring to a boil over high heat, then reduce heat and simmer until potato mashes easily when pressed, about 20 minutes.
6. In a blender, whirl soup, a portion at a time, until very smooth. Return to pan. Add shiitakes, wild rice with liquid, sherry, and half-and-half. Stir often over high heat until hot, about 2 minutes.
7. Ladle into bowls and season to taste with salt and pepper.