Preheat oven to 350. Salt, and bring two cups of water to a boil. Slowly sift in the polenta, stirring constantly until it begins to thicken, about 4-5 minutes. When it is the consistency of thick porridge, add olive oil and salt to taste. Cool slightly, then turn out onto a sheet of wax paper. Using a spoon, portion out the polenta into long sticks onto an oiled baking sheet. Bake for 6-7 minutes, or until crunchy. Set aside and keep warm.
Meanwhile, pour boiling water over mushrooms and cover to steam. Heat oil over medium heat in a non-stick skillet. Remove leaves from one sprig of rosemary, chop finely. Mince shallot. When the oil is hot, add a whole sprig of rosemary. Be careful, especially if the rosemary is wet – it could splash! Fry for a minute, flavoring the oil, then remove and discard. Add the shallots, cooking about a minute and a half to soften. Drain the mushrooms, reserving 2-3 tablespoon of broth. Add the broth and red wine to the skillet, reducing for 3-4 minutes. Roughly chop the mushrooms, adding them to the skillet with the remaining chopped rosemary. Cook for another 4-5 minutes until the flavors have mended together and the mushroom mixture has lessened in volume. Adjust salt to taste and serve with polenta sticks.