To rehydrate porcini mushrooms, place in a small bowl and add 3/4 cup boiling water. Soak for 30 minutes. Strain and reserve soaking liquid
In a large skillet, heat olive oil over medium heat. Add onion, carrot, and celery; season with salt and pepper; cook, stirring occasionally, until vegetables begin to soften, about 5 minutes.
Add garlic and cook for 1 minute. Add fresh mushrooms, rehydrated porcini mushrooms, and thyme sprigs; cook, stirring frequently, until mushrooms are almost tender, about 3 minutes. Add tomato and tamari, and cook, about 2 minutes more.
Add reserved porcini mushroom soaking liquid and wine. Simmer until thickened, about 20-25 minutes (NOTE: Prepare polenta while mushrooms are simmering). Add balsamic vinegar and reduce another 2-3 minutes. Add salt and pepper to taste. Remove from heat and set Mushroom Bolognese aside to cool.
Bring 4 cups water to boil in a medium saucepan over high heat. Add a pinch of salt to boiling water and slowly stir in polenta. Bring to a boil, stirring constantly, and immediately reduce to a simmer. Cook polenta, stirring occasionally, until no longer grainy, about 30 minutes. Remove from heat; stir in olive oil and season with salt.
Preheat oven to 350 degrees. Spoon half of the polenta into an 8” square pan or deep pie dish. Cover with half of mushroom Bolognese. Top with remaining polenta and another layer of mushroom Bolognese. Sprinkle top with cheese.
Bake until top is golden, about 25-30 minutes.