Heat 2 tablespoons olive oil in a large skillet over medium heat. Add onion and parsley stems; cook and stir until onion is translucent, about 5 minutes. Add zucchini and salt; cook and stir until softened, about 5 minutes.
Stir zucchini blossoms into the skillet. Reduce heat to medium-low, cover, and cook for 10 minutes. Uncover skillet and cook until moisture evaporates, 2 to 3 minutes more.
Whisk parsley leaves, eggs, Parmesan cheese, Pecorino Romano cheese, and mint together in a bowl. Fold in zucchini mixture from the skillet. Mix well; season with salt and pepper.
Heat remaining 1 tablespoon oil in a large skillet over high heat. Pour in egg mixture and spread evenly across the skillet. Cook for 2 minutes. Prick surface in several places with a fork. Reduce heat to low and cook for 5 minutes. Remove from heat and cover; let stand until top is just set, about 5 minutes.
Invert frittata onto a large plate and slide back into the skillet. Cook over low heat until second side is set, about 5 minutes more. Remove from heat and let stand 5 to 10 minutes before serving.
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