How to Make It
In a bowl, beat the eggs. Using a wooden spoon, mix in the flour and a pinch of salt until a smooth, thick pasta batter forms. Cover and let rest for 30 minutes.
Bring a large pot of salted water to a boil. Using a flexible spatula, carefully press 1/4 cup of the pasta batter through the holes of a colander into the water and cook until the pasta is firm and floats to the surface, about 30 seconds. Using a slotted spoon, transfer the pasta to an ice water bath to cool. Repeat to cook the remaining 3 batches of pasta. Drain well and pat dry.
In a skillet, melt the butter in the 2 tablespoons of olive oil. Add the zucchini rounds and season with salt and pepper. Cook over moderate heat, stirring, until softened, about 2 minutes. Stir in the pasta and season with salt and pepper. Cook, stirring, until warmed through, 3 minutes. Off the heat, stir in the zucchini flowers, if using, and basil. Garnish the pasta with cheese shavings, drizzle with oil and serve.