Cut any larger mushrooms in half. Heat half of the oil in a large frying pan, and add as many mushrooms as can be cooked in a single, non-crowded layer. Over medium heat, let the mushrooms cook without moving them for a few minutes. Working in batches, cook all of the mushrooms until soft and browned. As they're done, transfer them to a bowl with the butter, thyme, and garlic. Cover, and keep warm.
To cook the polenta, combine the polenta, water, rosemary, and tarragon in a saucepan. Cook over medium heat until thickened, stirring often. This should take about 25 minutes. When thickened, add half of the cheese and stir in. Season with salt & pepper to taste.
Pour the polenta out on an oven-safe dish, top with the mushrooms and remaining cheese. Broil for 4-5 minutes, or until the cheese is melted and bubbling.
Serve warm.
Recipe by The Kitchen Paper at http://c02.ddd.mwp.accessdomain.com/herbed-mushrooms-on-polenta-with-white-cheddar/