Cook the mussels: Heat a skillet large enough to hold the mussels in a single layer over high heat for 1 minute. Remove from the heat and add the mussels and 1/4 cup of the vermouth. Return the skillet to the stove and cook for 1 minute. Add about 3/4 cup 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 tongs to pluck the mussels from the pan as they open. When all of the mussels are open, remove them and strain the liquid, reserving it for the sauce. Remove the mussels from their shells. Discard the shells.
Cook the mushrooms: Rinse the skillet and return it to high heat until it smokes lightly. Remove the pan from the heat, add 1 tablespoon of the butter, and swirl the pan until the butter turns a light brown color. Return the skillet to the heat and immediately add the shallots and mushrooms. Season lightly with salt and the red pepper flakes. Cook the mushrooms until they start to give up liquid and brown, 3 to 4 minutes. Add the remaining 1/4 cup vermouth and cook for 1 minute. Season to taste with salt. Toast the hazelnuts: Meanwhile, in a small skillet, melt 1 tablespoon of the butter over medium heat and add the hazelnuts. Season them lightly with salt and toast them over low heat until they turn light brown, 5 to 7 minutes. Use a slotted spoon to remove the hazelnuts and transfer them to a cutting board. Roughly chop the hazelnuts.
Make the sauce: Transfer the mushrooms to a plate using a slotted spoon. If there is any cooking liquid from the mushrooms, combine it with the mussel cooking liquid in a small saucepan. Simmer over low heat 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 in 1 tablespoon of the butter and a touch of lemon juice. Keep warm.
Assemble the dish: Heat a large skillet over high heat until hot, 2 minutes. Remove from the heat and add the remaining 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. Stir in the sauce. Taste for seasoning. Toss in the hazelnuts. Serve immediately on a platter, family style, with slices of grilled bread.
Cleaning Mushrooms: While I was working at Guy Savoy in Paris, I learned to wipe each mushroom clean with a damp cloth. Mushrooms are sponges and submerging them in water, especially for a longer period of time, dilutes their flavor. For white mushrooms, I trim the very bottom of the stem (it's often coated in soil) and wipe the whole thing clean. If they are especially dirty? Rinse them briefly in cold water and then cut and cook them immediately after.