In a large pot, heat the oil over moderately low heat. Add the carrot, onion, celery, and garlic and cook, stirring occasionally, until the vegetables start to soften, about 5 minutes. Add the broth, rice, and salt. Bring to a boil. Reduce the heat and simmer, partially covered, for 10 minutes
Add the escarole to the pot. Bring to a simmer and cook until the escarole is almost tender, about 5 minutes. Add the chicken and pepper. Bring to a simmer and continue simmering for just 1 minute. The chicken should be done.
Remove the pot from the heat. Stir in the parsley and Parmesan.
For a richer soup, stir two well-beaten eggs into the pot along with the parsley and Parmesan cheese (after removing the pot from the heat).