In a medium saucepan, heat the olive oil until a piece of scallion sizzles in it. Season the tuna pieces with salt and pepper and carefully add them to the saucepan; they should be completely immersed. Cook over low heat at a bare simmer until the tuna is just cooked through and a fork inserted in the fish comes out easily, about 7 minutes. Transfer the tuna to a plate and cover with foil to keep warm.
In a small saucepan, heat 3 tablespoons of the tuna cooking oil. Add the scallions and ginger and sauté over low heat until the scallions are tender, about 10 minutes. Add the balsamic vinegar and simmer for 1 minute. Stir in the soy sauce and ketchup and simmer gently for 1 minute. Remove from the heat and stir in the butter until smooth. Season the sauce with salt and pepper.
Set 2 pieces of tuna on each plate. Using a slotted spoon, spread a layer of the scallions on top of the tuna. Spoon the sauce on the fish and serve.