Preheat oven to 200°F. Sprinkle 1 side of tuna steaks with pepper. Heat 2 tablespoons oil in heavy large skillet over high heat. Place tuna steaks, pepper side down, in hot oil and sear 2 minutes. Turn tuna over and continue cooking to desired doneness, about 2 minutes for rare. Transfer tuna to rimmed baking sheet; keep warm in oven.
Add butter, sliced green onions, cilantro, ginger and chopped garlic to same skillet and saute until fragrant, about 30 seconds. Mix in mushrooms and soy sauce and simmer 30 seconds. Add whipping cream and simmer until sauce lightly coats back of spoon, about 3 minutes. Stir in lime juice.
Spoon sauce onto plates; arrange tuna atop sauce. Garnish with lime wedges and cilantro sprigs, if desired.