In a heavy bottom Dutch Oven pan, heat 2 tablespoons olive oil over medium high heat.
Add the mushrooms and black pepper. Cook for 5 minutes. Mushrooms will reduce to less than 1/2. Add 1/4 teaspoon salt - stir. Remove mushrooms from pan and set aside.
Add 1 tablespoon olive oil. Reduce heat to medium. Add shallots and sauté for 1 to 2 minutes. Add garlic and thyme and sauté for about 1 minute until fragrant.
Add wine. Cook until reduced to half. Add chicken broth. Bring to a boil.While waiting for the broth to boil, in a small bowl, combine the beaten eggs and parmesan cheese. Set aside.
Add polenta and 1 teaspoon salt. Use a whisk to mix constantly to prevent lumps. Cook for 3 minutes. Add the mushrooms and continue cooking for 2 more minutes - whisk or use a spoon. Add the butter if using. Whisk until incorporated.
Turn off the heat. Add the egg and cheese mixture and mix to thoroughly combine.