Preheat oven to 425°.
Rub fillets with 1 1/2 teaspoons olive oil; sprinkle with 1/4 teaspoon salt and 1/4 teaspoon pepper.
Rinse fennel thoroughly. Trim stalks to within 1 inch of bulb. Discard hard outside stalks; chop 1 tablespoon fennel fronds, and set aside. Reserve remaining fennel fronds for garnish, if desired. Cut out tough core from bottom of bulb. Starting at 1 side, cut bulb horizontally into 4 (1/2-inchthick) slices.
Heat remaining 1 1/2 teaspoons oil in a large ovenproof skillet over medium-high heat. Add fennel slices, and cook 3 minutes on each side or until browned (slices may separate). Pour orange juice and mirin over fennel, and sprinkle with remaining 1/2 teaspoon salt and remaining 1/4 teaspoon pepper. Bring to a boil; reduce heat, and simmer, uncovered, 10 minutes (if fennel slices separate, arrange pieces in 4 circles). Place fillets on top of fennel slices; spoon orange juice mixture over fillets. Bake at 425° for 10 minutes or until fish flakes easily when tested with a fork.
While fish cooks, combine chopped fennel fronds, butter, lime juice, and chopped garlic in a bowl; stir well.
Using a spatula, gently lift fennel and fish out of pan, and place in wide, shallow bowls. Spoon pan juices over fish, and top with fennel butter.