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Cut off the grape-leaf stems and any tough center ribs, keeping the leaves intact. Select the 24 largest leaves. If necessary, fit 2 smaller leaves together to make 1 large one.
In a small saucepan, cover the quinoa with the water and bring to a boil. Add a pinch of salt, cover and cook over low heat for 12 minutes. Remove from the heat and let stand, covered, for 5 minutes. Fluff the quinoa.
In a small saucepan of boiling water, simmer the shrimp until cooked through, about 1 minute. With a slotted spoon, transfer them to a plate to cool, then cut the shrimp in half lengthwise. Add the beet to the water and simmer until tender, about 20 minutes; add more water if necessary to keep the beet covered. Drain the beet and let cool, then peel and cut it into 1/4-inch dice.
Put the capers in a bowl, cover with water and soak for 3 minutes, then drain and rinse. Coarsely chop the capers.
In a small skillet, toast the cumin seeds over moderately high heat until fragrant, about 30 seconds. Transfer the seeds to a work surface to cool, then finely chop. Put the cumin seeds in a bowl and stir in the capers, parsley, lemon juice, oil, garlic and lemon zest.
In a large bowl, combine the quinoa, shrimp and beet. Add the dressing, season with salt and pepper and mix well.
Spread 6 grape leaves on a work surface. Place a rounded tablespoon of the quinoa filling near the stem end of each leaf. Fold the bottom of the leaves up and over the filling, then roll them up, pressing firmly and folding in the sides. Transfer the rolls to a large platter, seam side down. Repeat with the remaining grape leaves and filling. Refrigerate the rolls, but let them return to room temperature before serving.
The stuffed grape leaves can be refrigerated overnight.