To make the carrot and sweet potato puree, combine the vegetables in a saucepan and cover with water. Bring to the boil, then reduce the heat to a simmer and cook until soft when pierced with a fork or knife.
While the vegetables are cooking, combine the lime oil, ginger, spring onions and 1/2 teaspoon salt in a wide bowl.
Heat a frying pan over medium heat, and while it is heating, season your fish steaks with salt and pepper on one side (you may also need to cut your steaks into individual portions). Drizzle a little olive oil into the hot pan and add the fish steaks, seasoned side down. Once in the pan, season the top of the fish. Sear the fish pieces until you can see a line of cooked white flesh reaching about one-third of the way up the sides (the time depends on thickness), then flip the pieces and cook on the second side to the same point. This type of fish is best left a little pink in the middle, and it should still feel slightly bouncy when you press on the top. Transfer the grilled fish to the bowl of dressing, lightly tossing the pieces around in it. Leave to steep in the dressing for 10 minutes or so, flipping the pieces over occasionally.
While the fish is marinating, drain all the water from the cooked carrot and sweet potato. Add the butter, 1/2 teaspoon salt and some pepper and mash the vegetables until as smooth as possible. Add the lime juice and mash again. Taste for seasoning, adding more salt, pepper and lime juice as desired.
Spoon the mash onto warmed plates and top with the fish and sauce.