In a large stockpot, melt 1 tablespoon of butter over medium-high heat. Add the carrots and sauté for about three minutes. Add the celery and onion and sauté until all the vegetables are tender, about seven more minutes.
Add the chicken broth, lemon juice, bouillon, and white pepper. Bring pot to a boil, then reduce the heat and simmer for about 20 minutes.
In a small bowl, combine the remaining butter and the flour. Gradually whisk it into the soup mixture, being careful to whisk out any lumps. Simmer soup for an additional 10 minutes.
In another medium sized bowl, gently beat the egg yolks until they are a light yellow color. Slowly add some of the hot soup mixture to the eggs. Whisk constantly until eggs are combined. Then add the egg mixture to the pot, stir to combine, and heat through. Add rice and chicken.