Start with the mushrooms. Heat butter and oil in a skillet set over high heat. Add mushrooms and fry until lightly browned. Add onions and cook until they start to soften, stirring occasionally, over medium heat.
Add garlic and cook for another minute. Add cornstarch and toss until the mushrooms are evenly coated. Add beer and cook for 2 minutes.
Add broth, thyme, salt and pepper. Whisk continually until the gravy starts to boil. Simmer on low, stirring occasionally, for 5-10 minutes. Add salt and pepper to taste. If you prefer the gravy thinner add a little more water.
Meanwhile, make the polenta. In a medium saucepan, bring water and milk to a boil over medium heat. Slowly whisk in the polenta, stirring continually, about 3 minutes. Stir in the butter, parmesan cheese, salt and pepper. Turn off the heat, cover and let stand for 2 minutes. If the polenta thickens too much, thin with some additional milk.
Prepare the poached eggs. Fill a saucepan with ⅓ water, add a good splash of vinegar and bring to a boil. Break each egg separately in a coffee cup. Turn down the heat and keep the water just below the boiling point.
Pour the first egg very carefully into the water, start pouring as soon as the cup touches the water. Repeat this with the second egg, and cook on a very low heat for 4 minutes. Carefully fish out the poached eggs with a slotted spoon, place on a plate lined with kitchen paper. Cut away any loose egg whites.
Divide the parmesan polenta between bowls, spoon the sautéed mushrooms, top with poached eggs, and fried sage leaves. Sprinkle with chives, flaky salt and black pepper. Enjoy!