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Preheat oven to 400°F. Trim and discard the beets' tops and ends. Quarter the beets, pierce their tops with a fork, drizzle with olive oil, season with salt and pepper, then wrap in aluminum foil, place in a baking dish, and roast until tender, about 40 minutes depending on size. Cool, peel, and cut into bite-size pieces.
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Meanwhile, in a small pot over medium-high heat, combine the quinoa, 1 cup water, and a generous pinch of salt. Bring to a boil, cover, reduce the heat, and simmer until light and fluffy, about 15 minutes. Transfer the quinoa to a rimmed baking sheet, spread in one layer, and cool to room temperature.
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In a small bowl, combine 3 tablespoons of the olive oil and the lime juice, vinegar, shallot, maple syrup, 1/2 teaspoon salt, and a few grinds of pepper. Whisk until well combined.
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In a serving bowl, toss the beets, quinoa, pickled onion, tangerine segments, spinach, parsley, and sunflower seeds with just enough dressing to coat. Finish with feta and a few grinds of pepper and serve.