Combine chicken stock, water, chicken broth, and soup mix in a large pot over medium heat. Stir until all soup mix is dissolved, and bring to a boil.
Add carrots and celery, reduce heat, and simmer for 10 minutes. Add chicken and rice, cover, and simmer for 20 minutes. Add half-and-half and pepper and simmer for 10 more minutes, being very careful it doesn't scorch.
Meanwhile, melt butter in a small pot. Add flour and stir to make a paste. Cook for 1 minute, then slowly add 2 ladles of liquid from the soup; whisk until no lumps remain. Add this mixture to the soup and stir until well combined. Continue to simmer until soup thickens slightly, 2 to 3 minutes. Serve immediately.