1. Preheat oven to 400 degrees.
2. In a large skillet, heat butter until melted. Add onion and garlic. Sauté over medium heat while stirring evenly for 20 minutes, or until lightly caramelized.
3. Remove pan from heat, and add balsamic vinegar and olive oil. Season with salt and pepper to taste. Set aside.
4. Line a baking sheet with parchment paper. Cut puff pastry into four 4" rectangles for ideal appetizer serving and place them onto the baking sheet.
5. Take the onion and garlic mixture and spread evenly (about ¼ cup) into the center of each pastry rectangle. Leave a 1" border around the edges.
6. Place a few slices of tomato atop the onion mixture and season with salt, pepper, and oregano to taste.
7. Place three anchovy fillets on top of each pastry.
8. With the remaining 1" of pastry, fold the edges up to meet the filling and crimp them with a fork.
9. Brush the dough with the beaten egg and move the pan to the oven.
10. Bake for 20–25 minutes or until golden-brown.