Step 1

In a small bowl, whisk together the whole egg plus yolk, both cheeses, the lemon zest and ½ teaspoon black pepper; set aside. In a 12-inch nonstick skillet over medium, combine 1½ tablespoons of the oil and the garlic. Cook, stirring occasionally, until the garlic is lightly browned, 2 to 3 minutes. Remove and discard the garlic, then add half the zucchini in an even layer and sprinkle lightly with salt. Cook, without stirring, until golden brown on the bottoms, 3 to 4 minutes.

Step 2

Stir the zucchini, flipping the slices, then redistribute in an even layer. Cook until golden brown on the second sides, another 3 to 4 minutes. Transfer to a plate. Add the remaining 1½ tablespoons oil to the skillet, then cook the remaining zucchini in the same way. Off heat, add the first batch of zucchini to second in the skillet. Stir in the pepper flakes and set aside.

Step 3

In a large pot, bring 4 quarts water to a boil. Stir in the pasta and 1 tablespoon salt, then cook, stirring occasionally, until al dente. Reserve about 1 cup of the cooking water, then drain. Return the pasta to the pot, then add ½ cup of the reserved pasta water and the zucchini. Cook over medium-high, stirring, until heated through, about 1 minute.

Step 4

Off heat, add the egg mixture; stir until the sauce thickens slightly and clings to the pasta and zucchini, 2 to 3 minutes. If needed, add reserved pasta water 1 tablespoon at a time to adjust the consistency. Taste and season with salt and black pepper. Serve sprinkled with additional black pepper and cheeses.