Set oven rack in the middle of the oven, and preheat to 425˚F.
Wrap the beets tightly in aluminum foil or parchment packets of 2 or 3 beets, and roast in the center of the oven for 40-45 minutes, until tender. Unwrap and allow to cool.
Toast the sunflower seeds in a dry skillet over medium heat, stirring occasionally, until golden and fragrant. Remove from pan and set aside.
In a small bowl, whisk together the vinegar, salt, pepper, sugar, and shallot. While whisking, slowly add the olive oil and whisk until emulsified.
When the beets are cool enough to handle, carefully remove the skins and discard. Cut into slices 1/4-inch thick, and set in a medium mixing bowl. Gently toss with half the vinaigrette. Arrange the beets on a serving platter. Combine the sunflower greens and sunflower seeds in the mixing bowl and toss with the remaining vinaigrette. Top the beets with the dressed sprouts, seeds, and chevre (if using), and serve.