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Heat 1 tablespoon oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add zucchini and cook until well browned, 5 to 8 minutes; transfer to bowl.
Add eggplant, remaining 2 tablespoons oil, ½ teaspoon salt, and ⅛ teaspoon pepper to now-empty skillet and cook over medium-high heat until eggplant is well-browned, 6 to 9 minutes. Stir in tomatoes, water, herbes de Provence, remaining ¼ teaspoon salt, and remaining ⅛ teaspoon pepper. Simmer until eggplant and tomatoes are softened, 7 to 10 minutes. Stir in zucchini and any accumulated juices and the vinegar.
Off heat, using back of spoon, make 4 shallow 1½-inch indentations in ratatouille. Crack 1 egg into each indentation. Spoon sauce over edges of egg whites so whites are partially covered and yolks are exposed.
Bring to simmer over medium heat (there should be small bubbles across entire surface). Reduce heat to maintain simmer. Cover and cook until whites are just set and yolks are still runny, 4 to 7 minutes, rotating skillet occasionally for even cooking. Sprinkle with Parmesan and basil and drizzle with extra oil. Serve.