Fold 4 large square sheets of parchment paper in half. Starting at the fold, cut out a large half heart shape. Open the paper flat on a surface with the point of the heart towards you. Butter the centers of each heart lightly with a teaspoon of the butter.
Divide the Swiss chard into 4 portions. Place a pile onto each heart, centering it on 1 side of the heart, towards the crease. Top each with 1/4 of the zucchini, asparagus, corn, and mushrooms. Season the fish well with salt and pepper and place a fillet on top of each pile of vegetables. Place 1 teaspoon of butter on each fillet and sprinkle each with some of the parsley, tarragon, shallots, and garlic. Drizzle a little white wine over each.
Fold the paper over to cover the fish. Starting at the top of the heart, tightly fold over the edges, overlapping each time, all the way to the point, until it is sealed. Tuck under the last fold to keep it closed. Butter the tops of the packets with the remaining butter. Place the packages onto a baking sheet and put them into the oven for 15 minutes.
To serve, put the papillotes onto plates and unfold them at the table. (Be careful of the hot steam.) Serve with lemon wedges.