Soak mushrooms in boiling water for 30 minutes. Drain.Cut off and discard mushroom stems.
Slice mushroom caps finely and set aside.
In a heavy, medium-sized non stick skillet over medium heat, saute ginger and garlic in oil for 30 seconds.
Add button mushrooms and saute until almost dry, about 8 minutes.
Add shitake mushrooms and cook 1 minutes Remove from heat.
Stir in soy sauce and sesame oil. ( Maybe prepared 1 day ahead. Cover and refrigerate.) Set aside.
Place 1 fillet flat side down on work surface. Place 2 tablespoons filling across widest short end and roll into a cylinder.
Repeat until all fillets are filled and rolled. Arrange fillets seam side down in 9 inch glass pie dish. ( May be done up to 6 hours ahead. Cover and chill. )
In a small bowl, whisk ingredients for sauce ( except carrot ) and any remaining mushroom filling. Spoon over fish.
Cook fish in pie dish in covered steamer over medium heat until fish is cooked through, about 12 minutes.
Transfer fish to plates. Spoon sauce over fish. Garnish with sliced carrots.