In a medium saucepan, heat the first tablespoon of olive oil. Add the shallot and sauté for a few minutes, until the shallot is translucent. Add the quinoa and cook and stir about 5 minutes.
Stir in the vegetable broth and salt, bring to a boil, reduce heat and simmer very low, covered, 30 to 35 minutes, or until the liquid is absorbed.
Meanwhile, pre-heat the grill to medium high. Trim the zucchini and cut it lengthwise, 1/4-inch thick. Toss in a bowl with the remaining olive oil and sprinkle generously with salt and pepper. Arrange on a vegetable grilling accessory (to prevent the vegetables from falling through the wider grates), and grill, turning the zucchini over, until slightly charred and wilted tender, yet still firm to the bite.
Toss the beet greens with their stems in the dressing. Add to the grill, and cook until the beet stems are just tender and the leafy greens have wilted and all but disappeared.
Coarsely chop the zucchini and beets and stir into the warm quinoa. Season to taste with salt and pepper, to taste.