Put the mushrooms in a bowl, pour a pint or two of boiling water over them and cover the bowl. Let this steep while you chop all the other vegetables and herbs.
Squeeze out most of the water in the rehydrated mushrooms (reserve the water), mince the mushrooms and pour the soaking water through a strainer with a paper towel set inside into another bowl. This removes any dirt or debris in the soaking water.
Heat the olive oil in a large pot over medium-high heat. When it's hot, add the onion and cook until it's turning a little brown on the edges. Add the mushrooms and garlic and cook another 4 or 5 minutes.
Mix in the tomato paste and herbs and let this cook, stirring often, for a couple minutes. You will have to work the tomato paste into the mushroom mixture.
Pour in the wine, turn the heat to high, let it come to a boil and let it boil a minute or two. Add the soaking water and stock. Mix well, add a little salt and let this simmer down, uncovered, for at least 1 hour. It will be very soupy at first. You want it to all cook down over time.
Once the sauce has reduced to a consistency you like, add salt if you need to, and grind fresh pepper over it. Serve with the polenta.
POLENTA
Polenta takes about 30 to 40 minutes to make, so you'll want to start before the mushroom ragu is totally done. Start by bringing 7 cups of water to a boil. Add salt, then stir in the polenta slowly in a steady stream, stirring constantly to avoid lumps. If you get lumps, use a whisk to break them up.
Turn the heat down to a simmer, and cover the pot. Stir the polenta frequently to prevent it from sticking. Depending on how coarse the polenta is, you will need 20 minutes to 40 minutes for it to become smooth and soft, not gritty.
When it's getting close -- and if it's too thick you can add a little water as you go, just not a lot at one time -- stir in the butter a tablespoon at a time, letting each one melt before adding the next. When you are ready to serve, mix in the grated cheese.
Notes
If you are in a hurry, you can halve the liquids so the sauce comes together faster.
Keys to Success
Time is your friend. The longer this cooks, the better it will be.
If you are using fresh mushrooms, add them with the onions.
If you happen to have it, add a tablespoon or two of porcini powder to the mix in the beginning. It adds a ton of flavor.
If you want meat in this, use 4 slices of bacon instead of olive oil, removing and chopping the bacon after it's rendered its fat. Eat some or add it back to the ragu at the end.
If you don't have pasta or polenta, this is great with crusty bread or mashed potatoes.