Soak the porcinis in 5 cups hot water, 30 minutes. Strain into a saucepan, squeezing the liquid from the porcinis; chop the porcinis and set aside.
Make the mushroom broth: Remove the shiitake stems and transfer to a separate saucepan. Add 5 cups water, the bay leaf, 1/2 cinnamon stick, the peppercorns and 1/2 teaspoon salt. Bring to a boil and cook 3 minutes, then remove from the heat and set aside to infuse, 30 minutes. Strain into the saucepan with the porcini soaking liquid, squeezing the liquid from the shiitake stems; discard the solids.
Meanwhile, chop the shiitake mushroom caps and mixed mushrooms. Heat 3 tablespoons butter and 1 tablespoon olive oil in an extra-large skillet over medium-high heat. Add the garlic, red pepper flakes and the remaining 1/2 cinnamon stick and cook, stirring, 1 minute. Add the shiitake caps and mixed mushrooms in an even layer and cook, undisturbed, until browned, about 5 minutes. Stir and continue cooking until the mushrooms are soft, about 3 more minutes. Season with salt and discard the cinnamon stick.
Bring the mushroom broth to a simmer. Heat 2 tablespoons butter and the remaining 1 tablespoon olive oil in a large saucepan over medium heat. Add the squash, onion, porcinis and 1/2 teaspoon salt; cook, stirring, 5 minutes. Add the rice and toast, stirring constantly, 2 minutes. Add the wine and simmer, stirring, until absorbed, 1 minute. Add 3 cups hot broth, return to a simmer and cook, stirring, until most of the liquid is absorbed, 8 to 10 more minutes. Add 3 more cups broth and the mushroom mixture; bring to a simmer and cook, stirring, until the rice is al dente, 10 to 15 minutes. Remove from the heat and stir in the remaining 1 tablespoon butter, the grated cheese and parsley. Add more broth, if needed. Season with salt and serve with more cheese.