Cook quinoa according to package directions. Spread in an even layer on a rimmed baking sheet to cool. Set aside.
In the bowl of a food processor, combine the pistachios, ½ cup basil, lemon zest, garlic, and extra-virgin olive oil. Pulse to create a coarse pesto.
Heat the avocado oil over medium heat in a large skillet with a lid. Add the beets and cover. Sauté, covered, for about 15 minutes, stirring occasionally, until beets are tender and beginning to turn golden brown. Uncover and set aside to cool.
Bring a large pot of water to boil then add the dandelion greens. Blanch for 2–3 minutes, until crisp-tender. Drain, run under cold water to stop cooking, and cool. Drain and spin-dry or dry well on a clean towel. Return the greens to the empty pot.
Add the cooled quinoa to the pot with the greens. Add the pesto and stir to combine. Spread the quinoa and greens in a shallow serving bowl and top with beets, goat cheese, and sliced basil. Serve at room temperature.