If you don't have a small cookie cutter, you can use a shot glass or egg cup to cut the puff pastry circles.
- Preheat oven to 400°F (200°C). On a lightly floured surface, roll out dough to a 1⁄8-in. (3 mm) thickness. Using a 2-in. (5 cm) round cookie cutter or glass, cut out 12 circles and place on a parchment paper-lined baking sheet. Beat yolk with water and brush over dough. Bake, 10 min.
- Remove baking sheet from oven and place a second baking sheet overtop, pressing down gently to stop pastry circles from rising any further. Place stacked baking sheets in oven and bake, 8 min. Remove from oven.
- Meanwhile, in a skillet over medium heat, cook butter and garlic without browning, about 2 min. Add sugar and vinegar, stirring until sugar is dissolved and liquid comes to a boil. Add tomato halves and cook 5 to 8 min. or until slightly caramelized. Season with salt and pepper. Remove top baking sheet and divide tomatoes over pastry circles. Bake, 5 min. Garnish with basil and cheese.