Place vegetable broth, rice, and thyme sprigs in the crock pot.
Cook on HIGH for 3 hours or on LOW for 6 hours.
In a mixing bowl, combine egg yolks and lemon juice; whisk until combined.
Remove thyme sprigs from soup.
Ladle 1 cup of the soup into the egg-yolk mixture; whisk vigorously until well combined. Make sure to whisk it very well, otherwise the eggs will cook and you will end up with scrambled eggs, instead.
Whisk the egg-yolk mixture into the crock pot; close and continue to cook for 10 minutes.
Remove lid and turn off the crock pot.
Stir in the heavy cream or milk.
Season with salt, pepper, and chili powder.
Garnish with parsley and lemon slices.
Serve.