Place a rack in the middle of the oven and preheat the broiler to high.
Bring a medium pot of water to a full rolling boil. Add the salt and then the noodles, turn off the heat, stir well, and let sit in the water for 8 to 12 minutes.
Stir from time to time. Check for doneness at about 8 minutes. You want the noodles to be al dente (soft but just slightly firm in the middle).
As soon as they are done, drain them through a colander and run cold water over them to stop the cooking. Set aside at room temperature.
In a small bowl, whisk together all of the ingredients for the shiitake seasoning.
Add the mushrooms and, with your hands, mix the seasoning onto the shiitakes.
Brush a little melted coconut oil onto a parchment paper–lined baking pan and add the shiitakes, gills down.
Broil for 5 to 6 minutes on one side, until browned, then flip over and broil on the other side for 3 to 4 minutes. Alternatively, grill over a medium-hot fire until caramelized on both sides.
To make the garnish, heat the oil in a small saucepan to 350°F.
Add the shallots and fry until lightly browned. Remove with a slotted spoon and drain on paper towels.
Sprinkle the salt over the top of the shallots and reserve at room temperature until ready to serve.
To serve, rewarm the mushrooms either in a hot oven or in a skillet, if necessary.
Place an equal amount of lettuce at the bottom of 4 bowls. Top with the noodles, carrot, cucumber, cabbage, peanuts, basil, mint, fried shallots, and warm mushrooms.
Serve the nuoc cham sauce at the table. Instruct your guests to apply liberally.