Move oven rack to middle position and preheat oven to 425 degrees F. Line baking sheet with parchment paper; set aside.
Cut pastry into two 9 ½ by 4 ½-inch rectangles. Place both on prepared baking sheet, and brush lightly with some of the beaten egg. Prick dough all over using a fork. Bake pastry for 10 minutes, then reduce oven temperature to 350 degrees F and continue to bake until pastry is golden brown and crisp, about 10-15 minutes longer. Cool pastry completely on baking sheet, about 30 minutes. Leave the oven on.
In medium bowl toss tomatoes with 1 tablespoon of olive oil, the vinegar, shallot, garlic, sugar, ½ teaspoon salt, and ¼ teaspoon pepper. Pour tomatoes into 13x9-inch baking dish and spread evenly. Roast tomatoes, without stirring, until slightly shriveled, about 30 minutes. Set tomatoes aside to cool.
Bring oven temperature back up to 425 degrees F. In small bowl mix together goat cheese with remaining 2 tablespoons olive oil until smooth, and season with salt and pepper to taste. Use tips of paring knife to cut ½-inch border around edge of cooled puff pastry. Gently press centers down so the top rests ¼ - ½ inch lower than the edges. Evenly spread goat cheese mixture over sunken shells. Spoon in roasted tomatoes evenly into each pastry shell.
Brush edges and sides of tarts with more of the beaten egg. Bake tarts until pastry is deep golden brown and tomatoes are heated through, about 10-15 minutes. Cool tarts on baking sheet for 5 minutes, then transfer to cutting board. Slice each tart crosswise into 3 pieces, sprinkle with basil and serve!