In a small bowl, soak the porcini in 1-1/2 cups warm water until softened, about 10 minutes. Lift the mushrooms out and reserve 1 cup of the liquid. Coarsely chop the mushrooms.
Meanwhile, in an 8-quart Dutch oven or other heavy-duty pot, heat 1 Tbs. of the butter with the olive oil over medium heat. Add the shallots and a pinch of salt and cook, stirring occasionally, until tender, about 4 minutes. Add the fresh mushrooms and 1 tsp. salt and cook, stirring occasionally, until the mushrooms start to soften, about 6 minutes. Add the porcini and cook until the mushrooms are very tender and release their liquid, 3 to 4 minutes. Add the vermouth and cook until all the liquid is evaporated, about 2 minutes. Add the reserved cup of porcini liquid, leaving any sediment behind in the measuring cup, and cook until reduced to about 3/4 cup, 3 to 4 minutes. Stir in the heavy cream and 3 Tbs. of the herbs. Season to taste with salt and pepper. Keep warm.
Bring 3-1/2 cups of water to a boil in a 4-quart heavy-duty saucepan over medium-high heat. Add 1-1/2 tsp. salt and then gradually whisk in the polenta. Cover, reduce the heat to low, and cook until the polenta absorbs the water and thickens, about 3 minutes. Stir in the remaining 2 Tbs. butter and the mascarpone. Add the Parmigiano and stir until combined. Serve the polenta topped with the mushroom ragoût and sprinkled with the remaining 1 Tbs. herbs.