How to Make It
Whisk together eggs and egg yolks in a small bowl until thoroughly blended. Stir in salt, pepper, and 1 cup of the cheese; set aside.
Heat oil in a large deep skillet over medium-high. Add ham and shallots, and cook, stirring often, until shallots are softened, about 6 minutes. Add peas and 3/4 cup water. Bring to a simmer over medium-high. Simmer, stirring occasionally, just until peas are tender, about 5 minutes. Remove skillet from heat.
Cook pasta according to package directions for al dente. Using tongs or a spider, transfer pasta to skillet. (It’s okay if pasta is still dripping water going into skillet.) Reserve 1 cup pasta water; let cool 2 minutes.
While whisking constantly, gradually pour the 1 cup reserved cooking water into egg mixture. Then slowly pour egg mixture into pasta mixture in skillet, stirring constantly with tongs.
Set butter and lemon juice in a small bowl alongside the stove. Place skillet over low heat, and cook, stirring and shaking skillet constantly, just until sauce is creamy and coats noodles, being careful not to overcook and curdle eggs, about 5 minutes. Remove from heat, and stir in butter and lemon juice. Divide pasta evenly among four warm bowls. Sprinkle evenly with chives and remaining 1/3 cup cheese.