Melt the butter over medium-low heat; add the mushrooms, shallots and parsley. Cook, stirring occasionally, until the juices evaporate and it is reduced in volume by half. Mash this mixture into the goat cheese with a couple of grinds of black pepper until well combined.
Stuff the blossoms with a teaspoon or two, depending on the size of the flower. Twist the open end to seal.
The batter is best if made at least an hour ahead of time so the starch can slowly absorb the liquid and become smoother and more cohesive. Sift together the cornstarch, flour and salt into a mixing bowl. Add the yolks and ice-cold water, stir to combine, cover and let rest until you are ready to use it. Just before dipping the flowers, whisk the reserved whites to soft peaks. Stir ⅓ of the whites into the batter to lighten the mixture, and then fold in the remaining whites with a rubber spatula retaining as much air as you can.
Heat ½ inch of vegetable oil in a Dutch oven over medium-high heat. A high-sided, heavy pot will prevent spatter and distribute the heat evenly. When hot—350° to 375°— dip the stuffed blossoms in the batter and then cook in the hot oil until golden brown on all sides. Cook them in batches without crowding, 5 or 6 at a time. They cook quickly, just a minute or so on the first side. Turn them over and cook a further minute until they have a nice golden color. Drain on absorbent paper and move to a wire cooling rack to keep them crisp while you cook the remaining blossoms.