Juice the lemons and set aside.
Mix eggs and cornstarch together in a medium size bowl until smooth.
Heat chicken broth until boiling, then add rice. Lower heat and cover pot to simmer until rice is soft, about 20 minutes. Turn off heat.
Beat egg mixture and milk together until blended.
Pour some hot broth into the egg/milk mixture to temper it, whisking constantly.
Add warmed egg/milk mixture to the chicken broth pot and continue whisking everything in the pot. Turn the burner on medium heat.
Heat until soup is thick, warm, and creamy but not boiling. Turn off heat.
Add butter, salt, pepper and lemon juice to the soup, blending thoroughly.
Serve immediately, or reheat until warm but not boiling.