Heat oven to 375°F. Line a 9×9-inch pan with parchment and lightly oil the paper and sides.
Grate zucchini, then squeeze out excess moisture with a clean towel until very dry.
Warm 1 tbsp olive oil in a skillet over medium heat. Soften onion for 3 minutes, then stir in garlic for 30 seconds.
Add spinach and cook just until wilted, 1 to 2 minutes. Set the mixture aside to cool slightly.
Whisk eggs, milk, salt, pepper, oregano, and baking powder in a large bowl until evenly combined.
Fold in mozzarella, Parmesan, dried zucchini, and the spinach-onion mixture.
Pour into the prepared pan and smooth the top. Bake 22 to 26 minutes, until set in the center and lightly golden.
Cool in the pan for 10 minutes. Lift out, slice into fingers, and serve warm or at room temperature.