Adjust an oven rack to the middle position and heat the oven to 425 degrees. Lay the puff pastry halves on a parchment paper–lined baking sheet and poke the surface all over with a fork to prevent uneven rising. Bake until the pastry is puffed and golden brown, about 15 minutes, rotating the baking sheet halfway through for even browning.

Using the tip of a paring knife, cut a half-inch-wide border around the top edge of each hot pastry, then gently press the centers down with your fingertips to create a well for the filling.

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

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

Whisk the goat cheese, most of the basil, the remaining oil, lemon zest, and pepper together in a small bowl until smooth. Spread the cheese mixture evenly over the centers of the par-baked pastry shells, then spoon the fennel mixture over the top.

Return the tarts to the oven and bake 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 warm.