Adjust an oven rack to the middle position and heat the oven to 425°F. Lay the puff pastry halves on a parchment paper–lined baking sheet and poke all over with a fork to prevent them from rising too unevenly.

Bake the pastry until puffed and golden brown, about 15 minutes, rotating the baking sheet halfway through for an even color. Using the tip of a paring knife, cut a half-inch-wide border around the top edge of each pastry, then gently press the centers down with your fingertips to create a shallow shell.

Heat oil in a 12-inch skillet over medium-high heat until shimmering. Add the fennel and cook until softened and browned, about 10 minutes. Stir in the garlic and cook until fragrant.

Stir in the wine, cover the skillet, and cook for 5 minutes. Remove the lid and continue cooking until the liquid has evaporated and the fennel is very soft, about 3 to 5 minutes. Remove from the heat, stir in the olives and lemon juice, and season with salt and pepper to taste.

Mix the goat cheese, basil, oil, lemon zest, and pepper together in a small bowl until smooth, then spread it evenly over the center of the pastry shells. Spoon the fennel mixture over the top. You can let the tarts sit at room temperature for up to 2 hours before the final bake if needed.

Bake the assembled tarts until the cheese is heated through and the crust is a deep golden brown, 5 to 7 minutes. Sprinkle with the remaining basil and serve while the pastry is still warm and crisp.