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.