Lightly season the chicken breasts with salt and pepper, then set them aside. Heat the olive oil in a large stock pot over medium-high. Sear the chicken breasts for approximately 2 minutes on each side, then transfer to a plate. Reduce the heat to medium and add the chopped onions to the pot, sautéing them for 3–4 minutes until softened. Incorporate the minced garlic, cooking for another 1–2 minutes until aromatic. Introduce the bay leaf and chicken stock paste, stir to mix, then pour in the water. Bring the mixture to a boil. Once it reaches a simmer, return the chicken breasts and any accumulated juices to the pot, then stir in the rice. Maintain a gentle simmer with the lid slightly ajar, cooking for 15–20 minutes until the chicken and rice are tender. Remove the chicken to a cutting board, keeping the soup warm by covering the pot. Shred the chicken using two forks and stir it back into the pot along with the chopped dill. Remove the soup from heat. Measure out 1 cup of the broth, avoiding solid ingredients, and set it aside.

To prepare the avgolemono, beat the egg whites in a bowl until foamy. Blend in the egg yolks, whisking steadily, then slowly incorporate the lemon juice while continuing to whisk. Gradually add the reserved broth to the egg mixture, whisking continuously to temper the eggs. Pour the finished avgolemono into the soup, stirring constantly to ensure even incorporation. Adjust seasoning with salt and pepper to taste. Serve the soup in bowls garnished with a sprinkle of fresh dill, a dash of salt and pepper, and a drizzle of olive oil.