Heat a skillet large enough to hold the mussels in a single layer. Add the mussels and 1/2 cup vermouth. Cook for 1 minute. Add about 3/4 cup cold water. The mussels shouldn't take long to cook, about 2 minutes. Take care to stir the mussels from time to time so they cook as evenly as possible. Use kitchen tongs to pluck the mussels from the pan as they open.
When all of the mussels are cooked, strain, and reserve the liquid for the sauce. Remove the mussels from their shells. Discard the shells. Reserve the mussels in the refrigerator until 5 minutes before serving.
Using a small paring knife, quarter the mushrooms. Heat another skillet large enough to hold the mushrooms in a single layer. Heat the skillet slightly and then remove the pan from direct heat. Add 1 tablespoon butter, melt quickly and turn a light brown color. Return the skillet to the heat and immediately add the shallot and the mushrooms. Season lightly with salt and red pepper flakes.
Add 1 tablespoon butter to a small skillet and add the hazelnuts. Season them lightly with salt and "simmer" them over low heat until they turn light brown. Set aside to cool.
Cook the mushrooms for 2 to 3 minutes, until they start to give up liquid. Add the remaining vermouth. Cook for an additional 1 minute and then drain out any cooking liquid. Combine the mushroom and the mussel liquids in a small pan and allow the 2 flavors to simmer together for 1 minute. The sauce may need a pinch of salt. The flavor should be an initial taste of sweet mussel followed by an undertone of earthiness from the mushroom. Swirl 1 tablespoon butter and a touch of lemon juice. Keep warm.
Heat the large skillet until hot, 2 minutes. Remove from the direct heat and add 1 tablespoon butter. When it turns light brown, return the skillet to the burner and add the mussels and mushrooms in a single layer. Cook for 1 minute without stirring them. They should be browned on their first side before you stir them slightly, with a wooden spoon, and leave them to brown for an additional minute. Taste for seasoning.
Toss in the hazelnuts. Serve on a platter, family style, with slices of grilled bread or over a piece of fish or meat for an aromatic and earthy celebration of mushrooms!
Wine pairing: A chalky chardonnay, Gruner Veltliner, or a pale ale would be lovely with this flavor combination.