In a large soup pot, melt the coconut oil. Add the onion and garlic and saute over medium heat until the onion is translucent.
Add the stock, water, salt, thyme, bay leaf, black pepper, and cayenne pepper. Bring to a boil.
Add the rice or orzo. Return to a boil. Cook until the pasta or rice is al dente.
Turn the heat down to medium-low. Add the shredded chicken.
In a medium bowl, whisk together the lemon juice and eggs. Pour some of the soup mixture into the egg mixture and briskly whisk to prevent coagulation. Add egg mixture to the pot.
Continue to heat, stirring often, until the soup thickens slightly.