Brush away any soil and grit from the mushrooms with a moist cloth, then slice the porcini and tear any other large mushrooms in half. Set aside.
Melt the butter in a large frying pan over a medium-high heat. Add the garlic, sage and the dense, tougher mushrooms and cook for 2–3 minutes. Add the remaining mushrooms, season with salt and pepper and cook for 2–3 minutes until soft and cooked through. Pour in the vermouth and cook, stirring, for 1 minute.
For the polenta, bring the measurement water to the boil in a large, heavy-based saucepan. Put the polenta in a jug and pour into the water in a slow but steady stream, stirring vigorously with a wooden spoon to prevent any lumps forming. Reduce the heat to a slow simmer and cook, stirring frequently, for about 5 minutes, or until the polenta is thick and comes away from the side of the pan. Stir in the butter and season with salt and pepper.
Divide the polenta between 4 serving plates, then top with the mushrooms.