Preheat the oven to 350°F (180°C).
Separate the beets from the greens and wash both well, scrubbing the beets with a brush. Place the whole beets on sheets of heavy-duty foil (if using a mix of varieties, use a separate sheet for each color). Drizzle each pile of beets with a few tablespoons of the vinaigrette and sprinkle with the thyme. Fold the foil up and over them to seal tightly.
Put the packets on a rimmed baking sheet and roast until the tip of a knife pierces a beet easily, 60 to 75 minutes. Let the beets cool in their packets.
While the beets are roasting, cut the beet greens and about 3 inches (7.5 cm) of the stems crosswise into thin strips.
In a medium saucepan, bring 1/2-inch (12 mm) water to a boil and add the greens. Cook until just wilted, 2 to 3 minutes, turning with tongs to ensure they cook evenly. Drain the greens and arrange them on a large serving platter.
Once the beets are cool enough to handle, cut off the pointed tips and stem ends, and cut the beets into wedges, leaving the skins on. Scatter them over the beet greens.
Sprinkle with the pomegranate seeds and crumble the goat cheese over everything. Drizzle with enough pomegranate vinaigrette to coat and garnish with the chives.