Preheat the oven to 350 degrees F.
Drain the sun-dried tomatoes in a sieve over a small skillet, letting the oil collect in the skillet. Roughly chop the tomatoes and reserve them in a small bowl. Heat the oil in the skillet over medium-high heat. Add the garlic and cook until golden brown, about 30 seconds, stirring occasionally. Add the thyme and let it crackle in the oil for about 10 seconds. Add the balsamic vinegar and season, to taste, with salt and pepper. Pour the warm oil over the reserved tomatoes. Stir to combine the ingredients well.
Line 2 baking sheets with parchment paper. Place 1 sheet of puff pasty on each baking sheet. Cut each sheet of puff pastry into 9 individual squares, each about 3 by 3 inches. For each square, use a paring knife to cut from each corner toward the center, stopping 1-inch from the center. You will have a 1-inch circle in the center of each square, and all of the corners will be cut in half. Spoon about 1 teaspoon of pesto in the center of each square and spread it into a 1-inch circle. For each square, lift one half of a corner and fold it toward the center of the square, gently pushing the point into the pesto. Continuing around the square, fold 3 more corner halves toward the center of the square, and gently press all 4 tips together to form a pinwheel shape. Draining away most of the oil, place about 1 teaspoon of the tomatoes where the tips meet. Brush the dough with the beaten egg. Bake at 350 degrees F until the dough is golden brown, about 20 minutes. Serve warm.