Remove the stems from the mushrooms. Finely chop enough mushroom stems to equal 1/3 cup. Discard the remaining mushroom stems.
In a large skillet, melt the butter over medium heat. Add the prosciutto, chives, and chopped mushroom stems. Cook, stirring frequently, until the prosciutto is crisp and the stems are tender, about 6 minutes. Reduce the heat to low. Add the cream cheese, mustard, lemon zest, tarragon, and pepper, stirring until the mixture is combined. Remove from the heat.
Spoon the cream cheese mixture evenly into the mushroom caps; place on a rimmed baking sheet. Bake 20 minutes. Serve immediately.
Mushrooms can be stuffed 2 days ahead. Cover and refrigerate.