Coat a loaf pan on all sides with cooking spray and set aside.
In a medium saucepan, bring water and salt to a boil. Slowly whisk in polenta. Reduce heat to low and simmer for about 5-10 minutes until mixture is thickened, whisking constantly. Season with pepper. Stir in cream and Parmigiano-Reggiano. Transfer polenta to loaf pan and smooth out top so mixture is evenly distributed in pan. Cover and refrigerate for at least 2 hours, or overnight.
Run a knife along inside of loaf pan to loosen polenta. Place a cutting board just large enough to cover loaf pan over its top and gently flip pan over to remove polenta. Cut polenta in half, then one half diagonal into two triangles (reserving 1 triangle and remaining half for a later use).
In a medium sauté pan, heat 1 tablespoon olive oil over medium heat. Add mushrooms and allow to cook, without stirring, for about 2 minutes. Stir and continue to cook for another 5 minutes, until softened. Add garlic and cook for another 1-2 minutes. Add thyme. Add wine and broth and simmer for about 2 minutes, until sauce cooks down a bit. Add butter and cook until incorporated.
In another sauté pan, heat remaining 1 tablespoon olive oil over medium-high heat. Fry polenta until golden brown, without touching, about 5 minutes. Carefully turn polenta over and continue to cook for another 5 minutes.
Place polenta on a serving plate and top with mushroom mixture, followed by additional Parmigiano-Reggiano. Serve immediately.