How to Make It
In a small bowl, soak the currants in the hot water. In a small saucepan, combine the honey, ginger and rosemary and simmer over low heat for 3 minutes. Remove from the heat.
In a large saucepan of boiling, salted water, cook the turnips until just tender, about 5 minutes. Using a slotted spoon, transfer to a shallow dish. Add the carrots to the boiling water and cook until tender, about 5 minutes. Drain.
Heat the olive oil in a large skillet. Add the turnips in an even layer and cook over moderately high heat for 1 minute. Reduce the heat to moderate and cook, undisturbed, until lightly browned on the bottom, about 4 minutes. Stir in the carrots and peppers and season with salt and pepper. Cover the vegetables and cook, stirring occasionally, until the peppers are tender, about 3 minutes.
Add the currants and their soaking liquid to the vegetables and cook until the liquid has thickened, about 2 minutes. Add the honey mixture, stir well and simmer for 1 minute. Season with salt and pepper and add the vinegar. Stir and transfer to a bowl. Serve hot or warm.