Preheat oven to 350°F. Place beets, 1 Tbs. vinegar, and 1 cup water in large baking dish. Season with salt, if desired, and bake 1 hour, or until beets are tender. Cool, then peel and quarter.
Bring quinoa and 2 & 1/2 cups water to a boil in medium saucepan. Cover, reduce heat to medium-low, and cook 20 minutes, or until quinoa is tender and water is absorbed. Drain, and cool. Toss with 1 & 1/2 Tbs. oil.
Mash avocado with tarragon, shallot, 1 Tbs. oil, lime juice, and lime zest. Season with salt and pepper, if desired.
Toss beets in remaining 1 & 1/2 Tbs. oil and remaining 1 Tbs. vinegar. Season with salt and pepper, if desired.
Divide arugula, beets, quinoa, and avocado mixture among serving plates. Sprinkle with pistachios, and garnish with grapefruit, if using.