Divide the beets and the onion into small groups, separating the vegetables by size. Wrap the groups tightly in aluminum foil and roast the beets and onion groups in the oven for 1 ½ to 2 hours, or until they are tender.
Unwrap the beets and the onions carefully and let them cook until they can be handled. Peel the beets and cut them into 1/3-inch-thick sticks. Peel the onions and cut them lengthwise into 1/3-inch-wide strips.
In a bowl toss the beets with the vinegar and salt and pepper to taste, add the onions and 2 tablespoons of the fresh thyme. Toss the mixture gently.
Transfer the salad to a serving bowl, sprinkle it with the remaining 1 tablespoon of thyme and serve the salad warm or chilled.