Prep the Mushrooms and Squash: Place the sliced shiitake mushrooms in a colander and toss with half a teaspoon of salt. Let them sit while you prep the squash to draw out excess water. Pat the mushroom caps dry with a paper towel just before cooking. Peel, seed, and cube the butternut squash into uniform ½-inch pieces so they cook evenly.
Sear the Mushrooms: Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Add the shiitake mushrooms in a single layer if possible (cook in batches if needed to avoid steaming). Cook undisturbed for 3-4 minutes until deeply browned on one side, then stir and cook until tender and fully browned, about 5 more minutes. Remove mushrooms from the pan and set aside on a plate.
Sauté Aromatics: Reduce heat to medium. Add another tablespoon of olive oil to the same skillet. Add the diced onion and a pinch of salt. Cook for 4-5 minutes until softened and translucent, scraping up any browned bits from the mushrooms. Add the garlic and dried thyme, cooking for just 30 seconds until fragrant.
Cook the Squash: Push the onions to the side of the pan. Add the remaining tablespoon of olive oil in the cleared space, then add the butternut squash cubes. Sear the squash for 2-3 minutes per side to develop color. Stir everything together and cook for another 5 minutes until the squash is fork-tender but still holds its shape.
Deglaze and Combine: Pour in the white wine (or broth) to deglaze the pan, scraping up any remaining fond from the bottom. Let it simmer for 1-2 minutes until reduced by half. Stir the cooked shiitakes back into the squash mixture. Add the balsamic vinegar and season with black pepper and additional salt if needed.
Serve: Taste and adjust seasoning. Serve immediately over creamy polenta or alongside crusty bread to soak up the flavorful juices. Garnish with fresh parsley if desired.
