Melt butter in a 9- to 10-inch nonstick skillet over moderate heat, then cook mushrooms and shallots, stirring occasionally, 3 minutes.
Add water and cook, stirring, until liquid is evaporated and mushrooms are tender and golden, about 5 minutes. Stir in tarragon.
While mushrooms are cooking, lightly beat together whole eggs, whites, parmesan, kosher salt, and pepper to taste.
Preheat broiler.
Add egg mixture to mushrooms, then cook over high heat, stirring briskly, until eggs are softly scrambled, 2 to 4 minutes. Reduce heat to low and form frittata into a 6 1/2-inch square with a spatula. Cook over low heat until bottom is set, about 2 minutes.
Place skillet under broiler (wrap nonmetal handles in foil), 5 to 6 inches from heat, and broil until eggs are just firm to the touch and barely golden, 1 to 2 minutes.
Transfer frittata to a cutting board and cool. Trim edges of square, then cut frittata into 24 rectangles.
Cut each prosciutto slice lengthwise into 3 strips. Wrap a prosciutto strip lengthwise around each egg rectangle, then tie a chive around crosswise.
Serve at room temperature.
Frittata squares can be made 2 hours ahead and chilled, covered with a dampened paper towel, then with plastic wrap. Bring to room temperature before serving.
Each serving (2 hors d'oeuvres) about 53 calories and 3 grams fat
