In a large skillet, heat 2 tablespoons of butter and 1 tablespoon of olive oil over medium heat. Once the butter is melted, add 8 ounces of mixed mushrooms and sauté until lightly browned, about 5 minutes. Then, add 1 chopped yellow onion and cook for an additional 3-4 minutes until softened. Stir in 2 minced garlic cloves and cook for 1 more minute.

Toss in 1 tablespoon of cornstarch and mix until fully combined. Pour in 1 cup of dark beer, stirring constantly for 2 minutes as it simmers and reduces. Next, add 1 cup of vegetable broth along with 1 teaspoon of dried thyme, salt, and pepper to taste. Let the mixture simmer for 5-10 minutes until it thickens.

In a medium saucepan, combine 2 cups of water and 1 cup of milk, bringing it to a gentle boil. Gradually whisk in 1 cup of polenta, stirring continuously for about 3 minutes until thick and smooth. Remove from heat, then mix in 2 tablespoons of butter, ½ cup of grated Parmesan cheese, and salt and pepper to taste. Cover and let sit for 2 minutes.

Prepare a small saucepan by bringing water with a splash of vinegar to a gentle boil. Reduce heat to simmering and carefully crack in 4 eggs, poaching them for about 4 minutes until the whites are set but yolks remain runny. Use a slotted spoon to gently remove the eggs and let them drain on a paper towel.

To serve, divide creamy polenta among bowls, creating a nest in the center for mushroom gravy. Spoon rich beer sautéed mushrooms generously over polenta, and gently place a poached egg on top. Finish with crispy fried sage leaves, a sprinkle of chopped chives, and a touch of flaky salt and pepper.