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.
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.
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.
To large pan, add remaining 1 tablespoon butter, onion, common mushrooms, garlic, rosemary, and thyme; stir often until vegetables are golden, about 10 minutes.
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.
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.
Ladle into bowls and season to taste with salt and pepper.