Bring a pot of lightly salted water to a boil. Add pasta and cook, stirring often, until al dente.
Meanwhile, beat eggs and yolks, cheese, olive oil, lemon zest and juice, pepper, and remaining 1 teaspoon salt together in bowl until well combined; set aside.
Reserve 1 cup cooking water, then drain pasta and immediately return it to pot.
Working quickly, whisk ⅓ cup reserved cooking water into egg mixture, then add egg mixture to pasta in pot while tossing pasta.
Continue to toss pasta until sauce begins to thicken and turns creamy, 1 to 2 minutes.
Add fresh dill and toss to combine.
Adjust consistency with remaining reserved cooking water, 1 tablespoon at a time, as needed, until sauce thickens to a creamy, silky consistency and leaves trails as you stir.
Season with salt and pepper to taste.
Serve immediately, passing extra cheese at the table.
