Set up your grill for indirect heat. If using a charcoal grill, build the coals on one side only. If using a gas grill, heat one side only.
Season the tomato halves with kosher salt and pepper and set aside. Place the sugar in a large enameled cast iron braiser or saute pan over the direct heat side of the grill. Shake out into an even layer and allow to melt then caramelize, 2 to 3 minutes. Add the vinegar and deglaze the pan, scraping the bottom of the pan for any brown bits. Stir in the garlic, onions and thyme, then the parsley and allow the liquid to melt back down. Remove from the heat.
Place the tomatoes flesh side down on top of the onions. Place on the indirect heat side of the grill, close the lid and vents of the grill (this will make the temperature drop to 225 to 250 degrees F) and bake until the skins are wrinkly and the tomatoes have softened and given up more liquid, 2 hours.
Brush another large enameled cast iron braiser or saute pan with the butter. When the tomatoes are ready, scoop the tomatoes out of their juices and place them in the buttered pan, cut side up and skin side down. Add the caramelized onions on top of the tomatoes. Using kitchen shears, cut off the square edges of the puff pastry, rounding the edges. Top the tomatoes and onions with the puff pastry round.
Place the pan on the indirect heat side of the grill, with the air vents open, and cover the grill. The temperature should be around 425 degrees F. Bake until golden brown and the pastry is puffed, 25 to 30 minutes. Cool in the pan for 3 minutes before turning out on to a plate. Garnish with a sprig of basil in the center and sprinkle with flakey salt.
If using an oven, roast the tomatoes at 225 degrees F for 2 hours. Bake at 425 degrees F with the puff pastry, 25 to 30 minutes.
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