Heat ¼ cup oil in a large Dutch oven over high. Add cremini mushrooms, and cook, undisturbed, until browned on one side, 8 to 10 minutes. Stir and continue cooking, stirring occasionally, until cremini mushrooms are browned all over, about 5 minutes. Transfer cremini mushrooms to a plate using a slotted spoon, reserving any oil in Dutch oven.

Add 3 tablespoons oil to Dutch oven over high. Add shiitake mushrooms; cook, undisturbed, until browned on one side, about 3 minutes. Stir and continue cooking, stirring occasionally, until shiitake mushrooms are browned all over, about 3 minutes. Transfer shiitake mushrooms to plate with cremini mushrooms; set aside.

Add remaining 2 tablespoons oil to Dutch oven over medium. Add onion, and cook, stirring often, until softened, 6 to 8 minutes. Add garlic; cook, stirring often, until fragrant, about 2 minutes. Add sherry, and cook, stirring often to loosen any browned bits on bottom of pan, until liquid is reduced by half, about 1 minute.

Add 5 cups vegetable broth, thyme, ground dried shiitake mushrooms (if using), salt, pepper, and 2 cups browned mushrooms to Dutch oven; reserve remaining about ½ cup browned mushrooms for garnish. Bring soup to a simmer over medium. Simmer, uncovered, until flavors meld, about 15 minutes. Remove from heat.

Transfer 3 cups of soup to a blender. Secure lid on blender, and remove center piece to allow steam to escape; place a clean towel over opening. Process until smooth, about 1 minute. Return pureed soup to Dutch oven with remaining soup. Stir in heavy cream and remaining 1 cup broth, ¼ cup at a time, until desired consistency is reached. Season to taste with salt. Divide soup evenly among serving bowls, and garnish with remaining browned mushrooms, thyme leaves, and black pepper.