What you are describing as arroz con pollo , to me is Fricase de pollo. I prepare mine with wine or beer. To the arroz con pollo I add beer( instead of water), chicken broth, cilantro, recao, ajo, cebollas, pimientos morrones,pimientos verdes, sofrito and a cubito knorr. After the rice is ready, I add green peas and asparragus for decoration and of course to eat. With the rice we have tostones or sweet plantains, red kidney beans, salad and sometimes bread. I ussually make all that when I have people over. For us is ussually the rice, beans plantains, and optional salad.
So your favorite dish is also Arroz con Pollo, I always watch my mother cook it and especially because she always cook some chicken as stew with potatoes and gravy. So when we serve ourself the rice we pour the chicken pieces and gravy on top of the rice.
Being from Puerto Rico we usually had sweet plantians or salad.
So do you have a special way of cooking it in your family. One secret I have is that cilantro is very good to add to the rice also.