1. Rinse tomatoes, core, and cut in half lengthwise. Trim off and discard fennel tops, root ends, and any bruised spots. Rinse heads and cut lengthwise into 1/4-inch slices. Peel and chop garlic.
2. Rub 1 tablespoon olive oil over bottom of 10- by 15-inch pan. Sprinkle garlic over half the pan and set tomatoes, cut side down and close together, on garlic. Lay fennel slices in other side of pan and drizzle with 1 tablespoon oil; mix gently with your hands to coat, then spread fennel evenly in half of pan.
3. Bake in a 350° oven until tomatoes are shriveled and fennel is translucent, about 1 hour (about 50 minutes in a convection oven). Let cool.
4. Meanwhile, rinse artichokes, drain, and trim discolored ends from stems. Put artichokes in a 4- or 5-quart pan. Add about 3 inches water, cover pan, and bring to a boil over high heat. Reduce heat and simmer until artichoke bottoms are tender when pierced, about 40 minutes.
5. Snap off and discard tough ends of asparagus. Rinse spears. Lift artichokes from pan and let cool. Add asparagus to water and turn heat to high. Cook, uncovered, until spears are tender-crisp when pierced, about 3 minutes. Drain and immerse at once in ice water until cold, about 2 minutes; drain again.
6. On a lightly floured board, roll each puff pastry shell into a 6- to 7-inch-wide round. Set rounds side by side on a 12- by 15-inch baking sheet; set another pan of the same size on top of the pastries.
7. Bake pastries in a 350° oven until well browned (lift top pan to check), 25 to 30 minutes (about 25 minutes in a convection oven).
8. As pastries bake, break leaves from artichokes and reserve for other uses. Scoop out and discard fuzzy centers from artichoke bottoms; also trim any coarse fibers from artichoke stems. Cut bottoms vertically into 1/8- to 1/4-inch-thick slices. In a bowl, mix with 1 tablespoon olive oil and vinegar.
9. Spread each warm pastry lightly with tapenade and set on a plate. Top each with 3 tomato halves and equal portions of fennel (don't wash pan; set aside), artichoke bottoms, and asparagus.
10. Top vegetables with equal portions of green salad (including dressing). Add salt and pepper to taste.
Salad: Rinse and crisp 4 cups (2 to 3 oz.) baby salad mix. In pan used to bake tomatoes and fennel, preceding, stir 2 tablespoons extra-virgin olive oil, 1 1/2 tablespoons balsamic vinegar, 1 teaspoon chopped fresh thyme leaves, and 1/2 teaspoon chopped fresh rosemary leaves, incorporating drippings in pan. Add salad mix and stir, seasoning to taste with salt and pepper.