Combine the rice, stock, and 2 cups of water in a big pot.
Cook over medium-high heat until bringing to a simmer. Turn to low heat. Cover the pot and leave a finger-wide a gap between the lid and the edge of the pot, to allow more steam to escape and prevent the congee from spilling or boiling over. Simmer for 25 minutes.
Add the chicken breast. Cook for another 15 minutes. Remove the chicken breast and place it onto a plate.
If the congee has reached the desired texture when the chicken breast is done, turn off the heat. Transfer the chicken onto a plate and shred it with two forks. If you want a stickier and thicker congee, remove the chicken first, then keep cooking the congee for another 15 to 20 minutes. Note, the congee will get quite thick and sticky towards the end. Stay nearby the pot and stir frequently.
When the congee reaches the desired texture, add back the shredded chicken. Add the century eggs, green onion, and sesame oil (if using) to the pot. Stir to mix well. Taste the congee and add salt to taste if needed.