Preheat oven to 400°. Line a rimmed baking sheet with parchment paper.
Lay 1 sheet of phyllo dough on prepared pan; brush lightly with oil. Top with another sheet of phyllo dough; brush lightly with oil. Repeat procedure until all phyllo sheets are used.
In a medium bowl, stir together cream cheese, goat cheese, and egg. Spread cheese mixture onto dough, leaving a 1-inch border on all sides. Top with sliced tomatoes and zucchini ribbons. Sprinkle with salt and pepper.
Bake until crust is golden brown and flaky, about 30 minutes. Let cool slightly; top with basil, oregano, and rosemary. Serve warm or at room temperature.