Preheat oven to 325 degrees and place racks in upper and lower thirds of oven. Combine Lion’s Mane mushrooms, zucchini and garlic on a large rimmed baking sheet. Season generously with salt and pepper; drizzle with oil. Toss to coat, then spread in an even layer. Place sheet on upper rack in oven and let mushrooms and zucchini roast while you prepare polenta.
Bring 4½ cups water to a simmer in a large ovenproof saucepan over medium-high heat. Add butter and a generous pinch of sea salt and whisk until butter is melted. Gradually add polenta, whisking constantly until combined. Return mixture to a boil, immediately cover pot, and transfer to lower rack in oven. While polenta is baking, occasionally shake baking sheet with mushrooms until polenta is tender, 25–30 minutes.
Remove polenta from oven. Turn up the oven heat to 400 degrees. Continue to cook mushrooms and zucchini until crisp around the edges, 5–10 minutes longer.
Meanwhile, carefully uncover polenta and whisk vigorously, scraping bottom of pan, until polenta is smooth and thick. Gradually add grated cheese if desired, whisking constantly until melted and incorporated; taste and season with more salt and pepper. Cover and keep warm over low heat while mushrooms finish roasting.
Remove mushrooms from oven; drizzle with balsamic; toss to coat.
Divide polenta among bowls and top with lions mane and zucchini and serve.