Combine first 9 ingredients in an 8-quart Dutch oven or stockpot; bring to a boil. Cover, reduce heat, and simmer 2 hours. Remove from heat.
Remove chicken and carrot from broth. Place chicken in a bowl; cover and chill. Dice carrot; cover and chill. Strain broth through a sieve into a large bowl; discard solids. Cover and chill broth at least 24 hours. Skim solidified fat from surface, and discard. Set aside 8 cups broth, reserving remaining broth for another use. Remove chicken from bones; cut into bite-size pieces. Discard bones.
Combine chicken, carrot, and 8 cups broth in Dutch oven; bring to a boil. Add pastina, Romano cheese, salt, and pepper; cook 5 minutes.
Combine egg and egg white; stir well, and slowly drizzle into boiling broth mixture, stirring constantly. Immediately remove from heat.