Season the chicken breasts with salt and pepper and set aside. In a large stock pot, add olive oil and turn to medium-high heat. Add the chicken breasts and sear for 2 minutes per side, then remove to a bowl or plate.

Turn the heat to medium and add the chopped onion. Cook for 3-4 minutes until softened. Add the minced garlic and cook for 1-2 minutes until fragrant.

Stir in the chicken stock paste and bay leaf. Add the water and stir to combine. Bring to a boil, then once simmering, add the chicken breasts (with any juices) and the rice. Stir and keep at a simmer at medium-low heat with the lid mostly on the pot, cooking for 15-20 minutes or until the chicken breasts and rice are cooked.

Once the chicken breasts and rice are cooked, turn the heat off and remove the chicken breasts to a cutting board. Leave the lid on the pot to keep the soup warm and use two forks to shred the chicken. Stir the shredded chicken back into the soup along with the fresh dill.

Remove the soup pot from the heat completely. Scoop out 1 cup of stock into a measuring cup and set aside.

Make the avgolemono: Separate the egg whites and yolks into two bowls. Whisk the egg whites vigorously until they develop a foamy consistency. Gradually add egg yolks to the egg whites and whisk to combine. Slowly pour in the lemon juice while whisking. Gradually pour in the 1 cup of stock into the egg mixture, whisking constantly to combine.

Add the egg mixture into the pot of soup, constantly stirring until combined. Taste and adjust seasoning with salt and pepper as desired. Serve in bowls topped with fresh dill, a pinch of salt and pepper, and a drizzle of olive oil.