Toss the apples with the lemon juice to coat in a small bowl.
Slice the sweet potato, using a Japanese mandolin slicer, lengthwise into 18 very thin slices. Adjust the mandolin to a thicker setting (matchstick thickness). Slice the remaining sweet potato into larger slices, and then slice those with a knife and cut into brunoise.
Bring a large pot of water to a boil and sprinkle with salt. Cook the sweet potato slices until just al dente, 3 to 5 minutes. Let cool on a sheet tray.
Heat the butter in a small saute pan over medium heat. Add the brunoise sweet potatoes and a pinch of salt and cook until tender but not mushy, about 5 minutes. Remove from the heat.
Mix the crushed red pepper flakes, scallions, half of the apples and the mustard in a large bowl and season with salt. Gently fold in the crab using a rubber spatula, being careful not to break up the meat.
On a clean flat surface, lay out 1 of the sweet potato slices. Spoon about 1 tablespoon of the crab filling 1 inch from a short side. Fold the border over the filling, and then continue to roll up the cannelloni. Transfer the cannelloni to plates seam-side down, using a small offset spatula. Continue the process, plating 3 cannelloni per plate. Top each plate of cannelloni with 2 teaspoons of the remaining apples and 2 teaspoons of the sauteed sweet potatoes. Cover with some of the Parmigano Sauce and a few sprinkles of grated Parmigano-Reggiano. Lightly brown the tops, using a torch, and serve with the arugula.
Heat the olive oil in a saute pan over medium heat. Once the oil is hot, cook the onions until softened, about 5 minutes, and then pour in the milk and simmer. Melt the butter in another saute pan over medium heat, and then add the flour and cook until the flour and butter resemble wet sand, 3 to 4 minutes. Strain the milk and gradually whisk into the butter and flour until incorporated, and then season with salt and nutmeg. Slowly bring the mixture to a boil, stirring frequently to prevent scorching on the bottom. Once the sauce has come to a boil and thickened, stir in the grated Parmigiano-Reggiano. Remove from the heat and stir in the creme fraiche. Taste and adjust the seasoning, if needed.
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