Bring the chicken broth to a boil in a medium saucepan over medium-high heat. Add the orzo and cook 2 minutes less than the label directs.
Meanwhile, whisk the egg, yolks, 3/4 teaspoon salt and 1/4 teaspoon pepper in a medium bowl. Whisk in the lemon juice. Reduce the heat under the orzo to low; scoop out 1 cup broth with a ladle and pour it into the egg mixture in a steady stream, whisking with the other hand. Then pour the egg mixture into the saucepan with the remaining broth and orzo in a steady stream, whisking constantly.
Stir the chicken into the soup. Increase the heat to medium and bring to a gentle simmer, stirring often. Cook, stirring, until the soup thickens slightly, about 4 minutes. Stir in the peas and carrots and warm through.
You need to warm up (or temper) your eggs before you add them to the broth or else you’ll get scrambled egg soup! Whisk the eggs with a little hot broth first, then slowly whisk them into the soup.