Bring a large pot of water to a boil. Fill a large bowl with ice water and set aside. Choose 9 largest chard leaves and trim stems flush with base. (Save remaining chard for other uses.) Thinly slice stems, cut 1 leaf into ribbons, and set both aside. Cook remaining leaves in boiling water 1 minute. With a slotted spoon, transfer leaves to ice water; let cool. Drain and spread out on kitchen towels.
Arrange 4 leaves horizontally on work surface, bumpy side up. Arrange a second leaf on top of each, perpendicular to the first.
In a medium bowl, combine 3 tbsp. oil, the parsley, 1 tbsp. thyme, minced garlic, the lemon zest, and 1/2 tsp. each salt and pepper. Add fish and turn to coat.
Set a fish piece at base of each chard set. Fold sides of leaves over fish, then roll to enclose. Brush all over with 1 tbsp. oil (total).
Heat remaining 2 tbsp. oil in a large, deep frying pan over medium-high heat. Sauté shallots until golden, 3 minutes. Add chopped garlic and sauté 1 minute. Add mushrooms, chard stems and ribbons, and remaining 1 tbsp. thyme and 1/2 tsp. each salt and pepper; sauté until mushrooms soften, 4 to 5 minutes. Stir in broth and bring to a simmer.
Set chard bundles in pan and reduce heat to medium. Cover tightly with lid or foil and cook until an instant-read thermometer inserted horizontally into center of fish reaches 115° to 120°, about 7 minutes. Transfer bundles to wide soup plates and spoon mushrooms and pan juices around them.