Scrub potatoes; cut each in half across widest dimension and brush cut sides with lemon grass-ginger butter, about 1 tablespoon total. Set potatoes cut side up in a shallow 3-quart casserole (about 9 by 13 in.).
Bake in a 400° regular or convection oven for 20 minutes. Turn potatoes cut side down and bake until well browned and tender when pierced, 20 to 30 minutes longer.
Meanwhile, in a small bowl, mix parmesan cheese, shallot, parsley, and garlic. When potatoes are done, sprinkle mixture evenly over them. Cover and keep warm.
Arrange crab legs in a single layer in an 11- by 17-inch pan. Brush with lemon grass-ginger butter, using about 2 tablespoons total. Broil 4 to 6 inches from heat until meat is hot in center of thickest part of leg (cut where cracked to test), 8 to 10 minutes.
Transfer crab legs to a platter, arrange potatoes around them, and sprinkle with the minced fresh basil. Serve with the remaining lemon grass-ginger butter to add to taste.