Adjust the oven rack to the middle position and heat the oven to 400°F. Line a large baking sheet with parchment paper.

Place the puff pastry halves on the prepared baking sheet and bake until the pastry is golden brown and puffed, 15 to 20 minutes.

Heat two tablespoons of olive oil in a large skillet over medium heat until shimmering. Add the fennel and salt; cook, stirring occasionally, until the fennel is softened and the edges are beginning to brown, about 8 to 10 minutes.

Stir in the garlic and cook until fragrant. Pour in the wine and simmer until the liquid has evaporated and the fennel is completely tender, about 5 minutes.

Remove the skillet from the heat and stir in the olives, lemon zest, and lemon juice. Season with salt and pepper to taste.

Whisk the goat cheese with the remaining tablespoon of olive oil and the basil in a small bowl until the mixture is smooth and easy to spread.

Use a paring knife to cut a shallow half-inch border around the edges of the baked pastry rectangles, then gently press down the centers to create a flat bed for the filling.

Spread the goat cheese mixture over the pressed centers and top evenly with the fennel mixture. Return the tarts to the oven and bake until the filling is hot and the pastry edges are a deep golden brown, about 5 minutes.

Serve warm, while the pastry is still light and flaky.