Preheat oven to 350 degrees F. Place puff pastry sheets on a floured work surface. Using a floured 2-inch cookie cutter, cut out 24 rounds. Place on a parchment-lined (baking paper-lined) baking sheet and cover with plastic wrap while cooking onion topping.
Warm oil in a heavy-bottomed frying pan over medium heat. Add onions and cook over medium-low heat until soft and slightly browned, about 10 minutes. Remove frying pan from heat and allow onions to cool.
Top each pastry round with onion mixture. Halve the anchovy fillets lengthwise and place two halves in a cross pattern on top of onions on each pastry round. Cut olives into slivers and place in angles of anchovy crosses. Bake until pastry is golden and crisp, 7 to 10 minutes.
Serve hot, garnished with basil leaves.
(Onions can be prepared 2 hours ahead, but pastry is best baked just before serving.)
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