To clean the morels, submerge them in a bowl of cold water, leave for 2 minutes, strain from the top, and repeat until the water that settles at the bottom is clear. Strain and pat dry with a towel. Melt the butter in a medium sauté pan over medium heat and add the shallot. Cook, stirring, until translucent, about 2 minutes. Add the morels, sprinkle with the salt, and cook, stirring occasionally, for 5 minutes. Add the vin jaune and simmer with the morels for 10 minutes. Scoop out the morels and set aside. Continue simmering until reduced to about 2 tablespoons. Add the chicken stock and simmer until the liquid has reduced by half. Add the chicken jus and simmer until the sauce coats the back of a spoon. Check the seasoning and add salt and pepper if necessary. Strain the sauce into a bowl set over ice; stir until chilled and reserve. Set aside 12 morels for garnish and finely chop the rest for the asparagus crust. Reserve, chilled.