Preheat oven to 350 degrees F (175 degrees C). Place pie crust on a pie pan or baking dish.
Heat olive oil in a skillet over medium heat. Cook and stir onion until softened and translucent, about 5 minutes. Reduce heat to medium-low, and continue cooking and stirring until the onion is very tender and dark brown, 15 to 20 minutes more.
Spread mozzarella cheese and tomatoes over the bottom of the pie crust. Whisk eggs, lemon juice, basil, salt, black pepper, and caramelized onion in a large bowl; pour mixture over cheese and tomatoes.
Bake in preheated oven until eggs are set and crust is flaky, 30 to 40 minutes.
