Bring 2 pots of water to a boil.
Meanwhile, peel the white radish, cut it into half lengthwise then cut across into 1-inch slices; also peel and slice the carrot.
Add 2 slices of ginger and 1 tsp sugar into the boiling water, then blanch the white radish for 5 minutes, reserve 1 cup of the water, and drain.
Rinse the duck legs and blanch them in boiling water for 3 minutes.
Trim any excess fat from the duck legs and cut it into 1-inch pieces.
Heat oil to medium hot, add the duck legs skin side down to brown slightly.
Then add the remaining ginger slices and saute with the duck legs till fragrant.
Turn heat to high and add the white radish and carrots, sauteing well.
Add the reserved water to about halfway, bring to a boil, then reduce heat to medium, cover, and cook for 15 minutes.
Remove lid, check if added salt is needed, then stir in the cornstarch paste a little at a time till sauce thickens, serve.