Preheat oven to 400 degrees. Measure (4) 24 in long sheets of parchment. Fold each sheet in half and starting from the folded side, cut a large half heart shape. (See step-by-step photos at usaweekend.com/recipes/fish en papillote)
Open heart on a work surface. Place one-quarter of the fennel a few inches from the center crease of the heart. Scatter one-quarter of the shallots over it. Place a fish fillet on top, parallel to the crease and season with salt and pepper. Scatter one-quarter of the mushrooms and fennel fronds over top, and, if desired, top with a pat of butter (you'll need a total of 4 tsps.) or a drizzle of olive oil.
Fold the paper over to enclose the fish and vegetables, and begin folding edges over: Make short folds, beginning at the deep center of the heart and working your way to the point of the heart. Each fold should overlap the previous fold; the folds will get gradually longer as you work your way around. After you fold the last pleat under, use a metal paper clip to secure it; you can also place a paper clip on the first fold to ensure a tight seal.
Refrigerate the packets; repeating with the remaining ingredients.
Place the packets on a baking sheet; bake for 15-18 mins or until the packets are puffy.