For the green beans: Bring 6 quarts of water to a boil in a large pot. When the water boils, add the salt to the water and then add the green beans. Cook for 4 minutes.
Prepare an ice bath: Fill a large bowl halfway with ice cubes and add some cold water. Place a colander squarely inside the ice bath. The colander will keep you from having to pick the beans out from amongst the ice cubes in the ice bath. When the beans are cooked, remove them from the hot water with a strainer and transfer them to the colander inside the ice bath. Use a spoon to swirl the beans around in the ice bath so they are sure to cool down immediately. Drain and set aside.
For the mushroom base: In a 10-inch cast iron skillet, melt the butter and add the mushrooms. Season with salt, to taste, and add the cayenne and mustard. Stir to blend and cook until the mushrooms give off most of their liquid, 5 to 8 minutes. Using a whisk, add the garlic and flour to the mushrooms. When all of the flour has been incorporated, add the chicken stock. Bring to a simmer. Taste for seasoning. Stir in the cream and sour cream and simmer gently. Continue cooking over low heat until the mixture thickens, 12 to 15 minutes.
For the shallots: Pour the oil into a medium frying pan. Heat to 350 degrees F.
In a medium bowl, combine the flour and cayenne. Toss the shallot rounds in the flour and shake off the excess by shaking the rounds in a strainer once they have been floured. Test the oil by dropping in 1 shallot slice. It should begin to bubble and fry gradually.
Prepare a baking sheet fitted with a kitchen towel. Drop a small batch of the shallots into the oil and gently swirl the oil as they fry. This will help them they fry more evenly on all sides. When they are light to medium brown, remove them with a slotted spoon and lay them out on the kitchen towel to cool. Sprinkle with a touch of salt.
To assemble: Stir the green beans and some of the shallots into the mushrooms. Simmer on the stove, over low heat until the green beans become tender when pierced with the tip of a knife, 10 to 15 minutes. When ready to serve, top with all of the remaining shallots.