Do not peel the onions. Slice a thin layer from the bottom and top of each onion. Scoop out the centers, leaving a skin, or shell, about 1/4 inch thick.
Chop enough of the onion centers to total about 1/3 cup. Brush the onions with oil.
In a large skillet, cook the 1/3 cup of reserved onion centers, mushrooms, carrot, and pepper in 1 tablespoon of butter for about 5 minutes.
Add the cider slowly. Stir in the chopped apple, the cooked rice, the nuts and salt, and continue cooking for another minute or so.
Remove from heat, stir in 1/2 cup of the bread crumbs, and the basil.
Spoon the onion/mushroom/rice filling into the onion shells. Arrange them in a baking dish and pour the remaining 1/3 cup of apple cider in the dish around the onions. Bake the onions, uncovered, in a 350 degree F oven for about 40 minutes, or until the onions are tender.
Melt the remaining butter and mix with the remaining bread crumbs, then sprinkle over the onions. Bake for another 10 or 15 minutes.
Remove the outer skins before eating.