Toss tomatoes with red onion, garlic, olive oil and vinegar and let sit, covered, for 30 minutes (or up to 3 hours in the fridge).
Strain tomato mixture, saving juices. Bring reserved juices up to a simmer and reduce to just 1 to 2 tbsp, then pour over tomatoes and toss (this concentrates the flavours).
Stir in basil and season to taste.
Preheat oven to 375°F and line a baking tray with parchment paper. Unfold puff pastry on a lightly floured surface (or if using a block, roll out to a square 8-x-8-inches and ¼-inch thick).
Cut puff pastry into 4 squares. Cut an L-shape on opposite corners of each square (creating a frame-like shape).
Lift one side of the cut and place it on the opposite side, and repeat with the other side (this creates a diamond-shaped shell that holds in the tomato filling).
Place on prepared baking tray. Spoon tomato filling into each tart shell and brush edges of pastry with egg mixture. Bake for 30 to 35 minutes, until pastry is a rich golden brown colour.
Crumble goat cheese over tarts and return to oven for 5 minutes.
Serves tarts warm or at room temperature.