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.
Mushroom Ingredient Note: Different mushrooms can be cut in different ways for bite-sized pieces. For creminis, cut in half. For maitakes (sometimes called hen-of-the-wood), break apart into bite-sized florets. For shiitakes, remove and discard stems and slice caps.
This nutritional information refers to each serving of the entire recipe and not just the products used as ingredients.