1. Adjust oven rack to middle position and heat oven to 425 degrees. Arrange puff pastry halves spaced evenly apart on parchment paper–lined rimmed baking sheet and poke pastry all over with fork. Bake pastry until puffed and golden brown, about 15 minutes, rotating sheet halfway through baking. Using tip of paring knife, cut 1/2-inch-wide border around top edge of each pastry, then press centers down with your fingertips.
2. While pastry bakes, heat 1 tablespoon oil in 12-inch skillet over medium-high heat until shimmering. Add fennel and cook until softened and browned, about 10 minutes. Stir in garlic and cook until fragrant, about 30 seconds. Add wine, cover, and cook for 5 minutes. Uncover and cook until liquid has evaporated and fennel is very soft, 3 to 5 minutes. Off heat, stir in olives and lemon juice and season with salt and pepper to taste.
3. Mix goat cheese, 1/4 cup basil, remaining 2 tablespoons oil, lemon zest, and 1/4 teaspoon pepper together in bowl, then spread evenly over center of pastry shells. Spoon fennel mixture over top.
4. Bake tarts until cheese is warmed through and crust is deep golden, 5 to 7 minutes. Sprinkle with remaining 1 tablespoon basil and serve.