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Cut bread into 1/4-inch cubes. Whisk eggs in a large bowl. Whisk in milk, wine (or another 1/2 cup milk), dry mustard, nutmeg, salt and pepper. Stir in the bread. Let stand at room temperature while you prepare the vegetables. (If baking the next day, cover and refrigerate the egg mixture.)
Heat oil in a large skillet over medium heat. Add onions, leek and garlic; cook, stirring frequently, for 5 minutes. Reduce heat to low and continue cooking very gently, stirring occasionally, until the onions have a little color and appear almost melted, 30 to 45 minutes more.
Meanwhile, bring 1 inch of water to a boil in a medium skillet. Add asparagus, cover and cook for 2 minutes. Drain and let cool; pat dry.
Coat a 9-by-13-inch baking dish (or similar 3-quart dish) with cooking spray.
When the onions are done, stir in scallions (or chives) and cook, stirring for 2 minutes. Transfer to the prepared baking dish and spread into an even layer. Sprinkle the asparagus over the onion mixture. (If baking the next day: cover and refrigerate the vegetables in the baking dish.)
If the egg mixture and vegetables were refrigerated overnight, remove from the refrigerator, uncover and let stand at room temperature for 1 hour.
Preheat oven to 350°F.
Scatter strips of prosciutto over the vegetables. Stir 1/4 cup cheese, parsley, mint and lemon zest (if using) into the egg mixture until well combined, then pour the egg mixture over the vegetables. Sprinkle the remaining 1/4 cup cheese over the top.
Bake the strata just until firm to the touch in the middle and an instant-read thermometer inserted into the center registers at least 160°F, 30 to 40 minutes. Remove from the oven, very carefully cover with foil to keep warm and let rest for 20 to 30 minutes before serving.