In a medium saucepan, cover the beets with cold water and bring to a boil. Simmer over moderate heat until tender when pierced, about 20 minutes for baby beets and 40 minutes for medium beets. Drain, then rinse in cold water and slip off the skins. Cut the baby beets in half lengthwise; thinly slice the medium beets crosswise. Let the beets cool.
In a large bowl, combine the vinegar with the shallots and season with salt and pepper. Gradually whisk in the oils until the dressing emulsifies. Add the greens, parsley, basil, tarragon, tomatoes and radishes to the bowl. Season the salad with salt and pepper, toss gently and transfer to a platter. Sprinkle the beets on top and serve.
The whole cooked beets can be refrigerated overnight.
A rich, round, powerful Champagne has enough body and fruity-toasty flavors to stand up to the aromatic herbs in the salad.