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Roast beets: Heat oven to 450 degrees. Place beets on a parchment-lined piece of aluminum foil and drizzle with 1 tablespoon olive oil. Wrap in foil and bake until beets are tender when pierced with the tip of a knife, 30 to 45 minutes for baby beets and up to 1 1/4 hours for larger beets. Let stand until cool enough to handle, then rub off skins with paper towels. Cut beets in half (or into quarters or sixths if large).
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Meanwhile, roast shallots, parsnips, and carrots: Toss shallots, parsnips, and carrots in a large bowl with remaining 3 tablespoons oil, then season with salt and pepper. Spread in a single layer on a parchment-lined rimmed baking sheet and sprinkle with rosemary, tossing to coat. Roast until tender and golden, turning vegetables over once, about 30 minutes. Transfer to a bowl.
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Prepare greens: Wash arugula and dry thoroughly, then place in a bowl and cover with a damp kitchen towel (or damp paper towels). Refrigerate until needed (this will help crisp the leaves).
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Toss vegetables and greens with vinaigrette: Toss parsnips, carrots, and shallots with 2 tablespoons vinaigrette. Toss beets separately with 1 tablespoon vinaigrette (to prevent their color from bleeding). Toss arugula with 2 tablespoons vinaigrette and season lightly with salt and a pinch of pepper.
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Compose salad and serve: Line a serving platter with arugula and arrange vegetables on top. Serve salad immediately.