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.
Cooks' note: Frittata squares can be made a day ahead and chilled, covered with a dampened paper towel, then with plastic wrap. Bring to room temperature before serving.
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