Rotate the poblanos over an open gas flame to char each side. Look for blistered, blackened skin around the entire pepper.
Enclose the charred peppers in a paper bag, fold the opening shut, and leave them for 15 minutes. The trapped steam loosens the skin from the flesh.
Rub away the skins, remove the stems, open the peppers, rinse out the seeds under warm running water, and cut the flesh into bite-sized pieces. The peeled pieces should retain their shape without papery skin attached.
During the pepper-steaming stage, heat the oven to 375°F and bake the chicken in a large casserole dish for 20 minutes, then distribute the prepared peppers over it and bake another 20 minutes. The chicken and peppers will release thin juices into the bottom of the dish.
Move the chicken and peppers into a large bowl, discard the liquid left behind, and use a fork to separate the chicken into large chunks. Keep recognizable pieces rather than pulling the meat into fine fibers.
Work the cream cheese, 1 cup of shredded cheese, heavy cream, and salt through the chicken mixture, then spread it back into the casserole dish. The dairy should form a smooth, thick coating around the chicken and peppers.
Cover the surface with the remaining shredded cheese and return the dish to the oven for 15 minutes, or until the casserole is hot and bubbling. Look for melted cheese across the top and active bubbling around the edges.
For an optional browned finish, broil the casserole for 1-2 minutes. Watch closely for golden patches on the cheese.