Set the instant pot to sauté.
Sauté the onion in the oil until the onion softens and becomes fragrant.
Add the carrots and celery. Cook for 30 seconds.
Add the chicken to the pot and stir.
Add the rice, salt, broth and lemon zest to the pot. Turn off the sauté function.
Place the lid on the pot and set the valve to sealing. Pressure cook for 5 minutes on high pressure.
Natural Release for 10 minutes (don't open the valve or the lid after it beeps). After 10 minutes, quick release.
If the rice soaked up more of the broth than you were expecting, just add more chicken stock to the pot.
Remove the chicken from the instant pot and break it up with two forks.
In a small bowl, mix the flour and butter. Add it gradually to the hot soup. Stir, and allow the soup to thicken.
Whisk the eggs in a medium bowl. Add juice of one lemon.
Scoop out around 1/2 cup of soup from the pot, and pour it SLOWLY into the egg/lemon mixture, whisking the entire time to prevent the eggs from scrambling. Repeat so you now have around 1 cup of soup mixed with the egg/lemon. Add this mixture back to the pot.
Add the chicken back to the pot.
Season the soup with salt and add more lemon juice if desired.
To serve, sprinkle with feta. If you don't love feta, then goat cheese or Parmesan cheese works nicely as well.