Cut mushrooms into bite-sized pieces. In a pot or Dutch oven heat 2 tablespoons oil over medium heat. Add half the mushrooms and cook until golden brown, about 4 minutes. Transfer mushrooms to a plate, add remaining 2 tablespoons oil to pan, and brown remaining mushrooms. Transfer to the plate.
With the oil left in the pan add shallots, carrot, salt, and pepper. Sauté until tender, 6-8 minutes. Add garlic and tomato paste and cook, stirring, 1 minute more.
Add mushrooms back to pan along with 2 cups broth, wine, thyme, and bay leaf. Bring to a boil, then reduce heat to simmer, partially covered, until mushrooms are very tender, about 20 minutes. Remove and discard thyme and bay leaf.
In a small bowl stir flour with 1 tablespoon water until smooth. Add flour mixture to mushrooms and simmer until thickened, about 5 minutes.
While mushrooms cook, combine prepared polenta with remaining 3/4 cup broth in a medium saucepan. Heat until liquid comes to a boil. With an immersion blender, puree polenta to make a creamy consistency. Season to taste with salt and pepper.
Serve mushrooms over polenta and garnish with chopped fresh parsley, if desired.
This nutritional information refers to each serving of the entire recipe and not just the products used as ingredients.