Preheat the oven to 350 degrees Fahrenheit.
Spread out the puff pastry on a piece of parchment paper and gently use a rolling pin to flatten out any creases. Cut into 16 equal rectangles: First, cut the pastry into 4 pieces, then into 8, and then 16. (If you're using another brand of puff pastry, your tartlets might be square rather than rectangular - either way works!) Using a small, sharp knife, score a smaller rectangle about 1/4 inch inside each of the 16 pieces. This might seem unnecessarily fussy, but it helps create and keep an edge on each of the tartlets.
Transfer the parchment paper with the puff pastry pieces to a baking sheet and nudge them away from each other a little bit so they aren't touching.
Spread each tartlet with 1/2 tablespoon of the pesto, staying within the boundaries of the scored inner rectangle. Place a few tomato halves on each tartlet, cut side up. Bake until the pastry is fully cooked and the edges are golden brown, about 30 minutes.
Sprinkle the tartlets with Grana Padano, drizzle with a few drops of olive oil per tartlet, and serve.
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