In a large saucepan, combine the chicken stock with 4 cups water, the carrot, celery, dill, shallot and 1/2 teaspoon salt. Bring to a boil over medium-high heat. Add the chicken breast, lower the heat to a bare simmer, and cook, turning occasionally, until just cooked through, 12 to 15 minutes. Transfer the breast to a plate. Remove the carrot, celery, dill and shallot from the broth, and discard.
Raise the heat to a brisk simmer. Add the orzo to the broth and simmer until tender, 6 to 8 minutes. Meanwhile, shred the chicken.
Whisk the eggs with the lemon juice in a medium bowl. When the orzo is cooked, slowly ladle about 1/2 cup of the hot broth into the egg mixture while whisking constantly. Pour back into the saucepan and cook, whisking occasionally, just until the broth returns to a simmer, about 2 minutes. Turn off the heat, add the shredded chicken, and season with salt.