Step 1Combine the basil, garlic, Parmesan, salt, pepper and oil in a food processor until a paste forms, 1 to 2 minutes. Adjust seasonings.
Step 1Heat the oven to 300 degrees. Combine the thyme, garlic, salt, pepper, sugar and olive oil. Spread over the bottom of a jellyroll pan.
Step 2Blanch the tomatoes in boiling water 30 seconds, then remove to a bowl of ice water. Slip off the skins and cut the tomatoes in half lengthwise. Place cut-side down on the pan; bake until softened, 1 1/2 hours. Remove from the oven.
Step 3Heat the oven to 400 degrees. Use a cutter 4 1/2 inches in diameter to cut 4 circles from the pastry. (You can also use a glass, or cut a 4 1/2-inch circle from cardboard.) Place on a baking sheet. Poke each a few times with a fork. Bake until golden and puffed, 12 to 15 minutes.
Step 4Place 4 (5-inch) squares of wax paper on a baking sheet. Place a 3 1/2-to 4-inch cake ring on each square. Using a slotted spoon, divide the tomatoes among the rings, skinned-side down. Spread 1 tablespoon of Pistou over each.
Step 5Place the tomato rings in the oven and bake until heated through, 10 minutes. Top each ring with a circle of pastry. Return to the oven and bake until heated through, 2 to 3 minutes. Slide a spatula under each tart, lightly hold the top, and turn over onto the center of a plate. Remove the wax paper and the rings (careful, they will be hot). Serve each tart surrounded by the greens.