Heat 2 tablespoons olive oil in a skillet over medium heat; cook and stir green onions in the hot oil until softened, about 3 minutes. Add garlic; cook until fragrant, about 1 minute. Remove from heat.
Combine goat cheese, Parmesan cheese, and cayenne pepper in a small bowl; stir with a fork until blended.
Gently flatten each puff pastry sheet with a rolling pin on a lightly floured surface. Cut each sheet into four squares. Lightly cut a border about 1/4-inch from the edges of each square using a sharp knife, not going all the way through the dough.
Preheat oven to 400 degrees F (200 degrees C).
Arrange puff pastry squares on a baking sheet; spread goat cheese mixture on each square. Top each square with onion mixture. Chill in refrigerator for 5 to 10 minutes.
Arrange zucchini and tomato slices in a pattern on each puff pastry square, allowing vegetables to overlap. Season with salt and black pepper. Drizzle with about 1 teaspoon olive oil.
Bake in the preheated oven until puff pastry is golden brown and puffed, about 20 minutes. Garnish with basil strips.