Peel yams and cut into 1/2-inch chunks.
In a 10- to 12-inch nonstick frying pan over high heat, combine yams, broth, and curry paste. Bring to a boil, then stir often until yams are tender when mashed, 10 to 15 minutes. With a slotted spoon, transfer to a bowl. Pour broth into another bowl. Wipe pan dry.
Meanwhile, rinse fish and pat dry. Return pan to heat and add fish. Cook until well browned on each side and opaque but still moist looking in center of thickest part (cut to test), about 10 minutes.
As fish cooks, mash yams with a mixer or potato masher. Add some of the reserved broth if you want softer yams. Save extra broth for other uses.
Spoon yams onto plates and top with fish. Scatter basil over portions and add salt to taste.