Heat a large nonstick skillet over medium-high heat. Add 1 tablespoon olive oil to pan, and swirl to coat. Add mushrooms and thyme; sauté for 4 minutes or until the mushrooms are tender. Add remaining 1 tablespoon olive oil, onion, bell pepper, garlic, 1/4 teaspoon salt, and 1/8 teaspoon black pepper; sauté for 8 minutes or until vegetables are tender.
Combine milk, 1/2 cup water, the remaining 1/4 teaspoon salt, and remaining 1/8 teaspoon black pepper in a medium saucepan; bring to a boil. Stir in polenta; cook 5 minutes, stirring frequently. Remove from heat, and stir in cheese. Serve with vegetables. Sprinkle with parsley.