Combine chicken stock, water, yellow onion, celery, carrots, tomato paste, salt, peppercorns, and bay leaf in a large pot; bring to a gentle simmer. Decrease heat to medium-low and simmer, checking flavor occasionally, until vegetables are slightly tender, 30 to 45 minutes.

Bring stock to a boil and add chicken, pouring in more water if chicken isn't completely covered in liquid. Bring poaching liquid to a boil; remove pot from heat and cover tightly with a lid until chicken is no longer pink in center, about 15 minutes. An instant-read thermometer inserted into center should read at least 165 degrees F (74 degrees C).