Place the whole chicken in a large pot with 2.5 litres of water. Add the halved onion, sliced carrots, and chopped celery. Season lightly with salt. Bring to a boil over medium heat, skimming any foam that rises to the surface to keep the broth clear.
Once boiling, reduce the heat to low and cover the pot partially. Let the chicken simmer for 1 to 1.5 hours, or until tender and fully cooked. This slow simmering extracts rich flavours from the chicken and vegetables, creating a comforting broth.
Carefully remove the cooked chicken from the pot and set it aside on a plate. Strain the broth, discarding the onion, celery, and carrot slices, and return the strained broth to the pot. Season the broth with salt and pepper to taste.
Add the rice to the broth and bring it back to a gentle boil. Reduce the heat to low and let the rice cook for about 15 minutes, or until tender. Stir occasionally to prevent the rice from sticking to the bottom of the pot.
While the rice is cooking, shred the cooked chicken meat into bite sized pieces, discarding the skin and bones. Reserve the shredded chicken for later, or add some back to the pot if desired.
In a medium bowl, whisk the eggs until frothy. Gradually whisk in the lemon juice, creating a smooth, slightly thickened mixture. This will be the base for the traditional Greek avgolemono (egg-lemon) sauce.
To temper the egg-lemon mixture, slowly ladle in about 1 cup of hot broth from the pot, whisking continuously. This prevents curdling by gently raising the temperature of the eggs. Once combined, pour the egg-lemon mixture into the soup, stirring gently to incorporate.
Add the shredded chicken back into the pot and warm through without letting the soup boil, as boiling can cause the eggs to separate. Serve the kotosoupa hot, garnished with fresh parsley. Accompany with crusty bread or pita on the side, and offer lemon wedges for an extra citrusy touch.