Preheat oven to 425 degrees. Chop the tops of the beats off (save the greens for another meal), scrub the beats, and then place beets on a large piece of foil on a baking sheet. Top with 1 tablespoon of extra virgin olive oil. Fold foil around the beets and crimp ends to form a packet. Roast beets in oven until tender (when pierced with a knife: about 45 minutes). Remove beets from foil and let cool preferably in the refrigerator. Once the beets are cooled they are very easy to peel. Once the beets are peeled cut into 1/4-inch-thick slices.
In a small bowl, whisk together remaining 3 teaspoons olive oil, thinly sliced shallot, lemon juice, and lemon zest and red wine vinegar; season with salt and pepper to taste. On a large platter or on individual plates, arrange beets and tomatoes (every other one being a beet) add 3 artichoke hearts per plate. Drizzle with dressing and sprinkle with thinly sliced basil leaves.
*You can refrigerate this salad up to 2 days; however, note that your tomatoes will become stained from the beet dye.