Add the chicken stock, bay leaf, and chicken breasts to a large pot. Bring just to a gentle simmer over medium heat and cook until the chicken is opaque and reaches 165°F at the center, 12–14 minutes. Transfer the chicken to a cutting board; discard the bay leaf. Pour the hot stock into a large heatproof bowl or pitcher and set aside.
Return the empty pot to medium heat. Add the olive oil and butter. Stir in the onion, carrot, and celery with a pinch of salt. Cook, stirring occasionally, until the vegetables are softened and lightly golden at the edges, 6–8 minutes. Add the garlic and red pepper flakes and cook 30 seconds until fragrant.
Stir in the orzo and toast it with the vegetables for 1 minute. Pour the reserved hot stock back into the pot and bring to a lively simmer. Cook, stirring occasionally, until the orzo is just tender, 7–8 minutes. Meanwhile, shred the chicken into bite-size pieces with two forks.
In a medium bowl, whisk together the egg yolks, whole egg, lemon juice, and lemon zest until completely smooth and slightly frothy. While whisking steadily, slowly stream in about 1 1/2 cups of the hot soup liquid to warm the mixture.
Reduce the soup to low heat. Stir the shredded chicken back into the pot. While stirring the soup in a slow circle, pour in the tempered lemon-egg mixture. Warm gently, 2–3 minutes, just until the broth turns creamy and lightly thickened—do not let it boil.
Season with the kosher salt and black pepper to taste. Stir in the chopped dill. Ladle into bowls and serve hot with extra lemon wedges if you like.