Preheat oven to 375 degrees. Wrap 1 sweet potato in parchment, and then in foil. Pierce several times with a fork. Bake until tender, about 1 hour. Let cool. Reduce oven temperature to 350 degrees.
Meanwhile, peel remaining sweet potatoes. Using a mandoline or a chef's knife, cut potatoes lengthwise into very thin slices until you have 30 slices. Trim each to a 2-by-4-inch rectangle.
Bring a large pot of water to a boil. Add half the sweet potato slices, and cook until tender, about 2 minutes. Using a slotted spoon, place slices on a baking sheet to cool slightly. Repeat.
Remove peel from baked sweet potato, and puree flesh in a food processor until smooth. Add cottage cheese, and puree until smooth.
Transfer potato-cheese mixture to a large bowl. Stir in apple, chives, grated Parmesan, salt, and pepper.
Coat a 9-by-13-inch baking dish with cooking spray. Place 1 heaping tablespoon filling in center of a sweet potato slice, and roll up. Place, seam side down, in dish. Repeat. (Cannelloni can be refrigerated, covered, overnight; bring to room temperature before baking.)
Brush oil over cannelloni. Bake until heated through, 10 to 15 minutes. Sprinkle with walnuts and shaved Parmesan.