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1. Wash brussels sprouts and trim roots. Separate leaves and place them in a bowl. Cover and refrigerate. (Can be completed the day before.)
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2. Warm a large saute pan over medium heat. Melt 2 tablespoons butter, then add onions and 1/2 teaspoon salt. Cook over medium-low heat, stirring frequently, until onions are soft, about 15 minutes.
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3. Peel, core and slice 5 apples. Add to pan with onions. Cook on medium heat, stirring occasionally, until apples soften and begin to break apart, about 20 minutes. Add apple juice and simmer for 10 minutes. Add mustard and remove from heat.
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4. Put apple-onion mixture in a blender or food processor and pulse 6 to 8 times until smooth. Keep warm.
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5. Warm a very large saute pan over medium heat. Melt remaining 2 tablespoons butter, then add shallots. Cook for 5 minutes, until shallots are soft and translucent. Stir in caraway seeds. Add brussels sprout leaves and cook over medium heat, stirring frequently, for 10 minutes. Add vinegar and continue cooking until leaves are just tender, about 5 more minutes. Season with salt to taste. Julienne remaining apple with skin on.
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6. Spoon apple-mustard puree onto a warmed serving platter. Pile sauteed brussels sprouts on puree. Sprinkle julienned apple on top. Serve immediately.
