1. In a large stainless-steel frying pan, heat 1 tablespoon of oil over moderate heat. Season the chicken breasts with 1/4 teaspoon salt and 1/8 teaspoon pepper and add to the pan. Cook until browned, about 5 minutes. Turn the chicken; add the broth, dill, and 1 1/4 teaspoons salt. Bring to a simmer, reduce the heat, and simmer, partially covered, until the chicken is just done, about 4 minutes. Remove the chicken and cover lightly with aluminum foil to keep warm. Set aside the pan with the broth.
2. Meanwhile, in a large pot of boiling, salted water, cook the orzo for 6 minutes. Add the carrots and continue cooking until the orzo and carrots are just done, about 6 minutes longer. Drain and toss with the remaining 1 tablespoon oil and 1/8 teaspoon each salt and pepper.
3. In a medium glass or stainless-steel bowl, beat the eggs, lemon juice, and 1/8 teaspoon of pepper until frothy. Bring the chicken broth back to a simmer and add to the eggs in a thin stream, whisking. Pour the mixture back into the pan and whisk over the lowest possible heat until the sauce begins to thicken, about 3 minutes. Do not let the sauce come to a simmer, or it may curdle. Put the orzo and carrots on plates and top with the chicken and sauce.
Wine Recommendation: Lemon and dill will work best with a full-flavored white wine with decent acidity. Try one from the southern part of Burgundy such as a Mâcon or Pouilly-Fuissé (both made from chardonnay grapes).