Preheat the oven to 375 degrees F.
Scrub the beet root bulbs under cold running water to remove dirt and debris.
Place the beets in a shallow baking dish and add the 2 tablespoons champagne vinegar, water, thyme sprigs, and salt.
Toss to coat the beets with the liquid.
Cover the pan with aluminum foil and roast for 45 to 60 minutes, until the beets are tender when pierced with a knife. Timing will depend on the size of your beets.
Transfer the pan to a wire rack and allow the beets to cool slightly (reserve the liquid in the pan).
When cool enough to handle, use a paper towel to remove the skin from the beets.
Cut the peeled beets into 1-inch pieces, return them to the remaining liquid in the baking pan, and toss to coat.
To make the vinaigrette, in a small bowl, whisk together the olive oil, 1 tablespoon of champagne vinegar, shallot, Dijon mustard, and honey (if using).
Season the vinaigrette with salt and pepper.
Transfer the arugula (or greens of choice) to individual salad plates.
Drizzle the vinaigrette over the greens.
Top the greens with the beets.
Tear or slice the burrata into smaller pieces and add to the salad.
Top the salad with the walnuts.
If desired, drizzle more vinaigrette over top.
Serve with extra vinaigrette on the side.