Mince the garlic clove and finely chop the fresh dill, setting both aside. Measure out the cooked rice, shredded chicken, and lemon juice into separate bowls for easy access. Pour the chicken broth into a large pot and bring it to a gentle boil over medium-high heat. Once boiling, season with salt and black pepper to taste—I prefer to taste as I go rather than add everything at once, as the broth will concentrate slightly as it simmers.
In a separate bowl, whisk together the 2 egg yolks, lemon juice, lemon zest, and 1 cup of the hot broth from the pot. Whisk continuously until well combined and smooth—this mixture is called a liaison and will add richness and brightness to the soup. Work slowly and deliberately to avoid curdling the eggs. I always add the hot broth gradually while whisking to temper the eggs, which prevents them from scrambling when added back to the hot pot.
Reduce the heat under the pot to medium-low. Slowly pour the egg-lemon mixture from Step 2 into the simmering broth, whisking constantly to distribute it evenly and prevent the eggs from curdling. Stir in the shredded chicken and cooked rice, then add the minced garlic. Simmer gently for 8-10 minutes, stirring occasionally, until the soup is heated through and the flavors have melded. Finally, stir in the fresh dill and taste for seasoning, adjusting salt and pepper as needed. Serve immediately in bowls while hot.