Preheat the oven to 425 degrees F.
Separately wrap the beets tightly in double layers of foil with the vinegar, 2 tablespoons olive oil and thyme leaves, and season well with salt and pepper. Roast in the middle of the oven until tender, 1 to 1 1/2 hours. Unwrap the beets and allow to cool.
To make the vinaigrette, whisk the lemon juice and shallots in a small bowl. Add the remaining olive oil in a steady stream. Season with salt and pepper.
When the beets are cool enough to handle, slip off the skins and discard. Separately cut the golden and red beets into 1/2-inch dice and put in separate bowls. Drizzle with the vinaigrette and season with salt and pepper.
Divide the beets onto 6 salad plates. Toss the arugula with the vinaigrette to coat and gently mound on top of the beets. Crumble the goat cheese on top. Serve immediately.
Recipe courtesy of Chef Lynn Crawford, 2009