Preheat the oven to 250 degrees F. Line a baking sheet with parchment paper.
Slice the ends off each tomato, cut them in half lengthwise and carefully remove the seeds. Put the tomatoes in a bowl, toss with the balsamic vinaigrette, and season with salt and pepper. Place the tomatoes cut side up on the baking sheet and roast for 3 hours. Transfer the tomatoes to a plate to cool. (Tomatoes can be made ahead and kept covered in the refrigerator for 2 days.)
Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper.
Put the artichoke bottoms into a bowl and toss with the lemon juice. Let sit for 5 minutes. Rinse the artichokes with cold water and pat them dry. Trim the bottoms so that they will sit flat and arrange on the baking sheet. Top each artichoke with half a roasted tomato, a mozzarella ball, and a teaspoon of pesto. Put them into the oven and roast for about 5 minutes. Serve immediately.
Oil packed sun-dried tomatoes may be used in lieu of oven-drying your own. Larger types of fresh mozzarella, cut into bite-size pieces, can be used in place of Ciliegini, or cherry tomato size mozzarella balls.