Drizzle beets with olive oil; wrap in parchment paper, then in foil. Roast on a rimmed baking sheet until tender, about 1 hour. When cool enough to handle, rub off skins, then thinly slice.
Meanwhile on a second rimmed baking sheet, toss edamame with 1 teaspoon of oil; season with salt and pepper. Roast, tossing halfway through, until golden, 20-25 minutes. Let cool slightly.
In a small bowl, whisk together vinegar and remaining 2 tablespoons olive oil. In a medium bowl, combine arugula, basil, beets and edamame. Toss with dressing, season with salt and pepper, and serve.
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