Preheat oven to 350°. In a medium pan over high heat, bring about 2 quarts water to a boil. Add asparagus and cook until bright green and barely tender, about 2 minutes. Drain and rinse with cold water.
Spread half the bread cubes in a lightly oiled 9- by 13-inch baking pan. Top with half the prosciutto, cooked asparagus, cheese, and chives. Layer remaining bread cubes on top, followed by remaining prosciutto, asparagus, cheese, and chives.
In a bowl, whisk eggs, milk, lemon peel, salt, and pepper to blend. Pour evenly over layered ingredients. Cover and chill at least 1 hour.
Bake until center of strata is set and top is lightly browned, 40 to 50 minutes. Serve warm or at room temperature.