In a blender or food processor, combine the peppers with stock, onions, garlic, cilantro, salt and pepper; puree. In a saucepot over medium heat, melt 3 tablespoons butter. Whisk in the flour; add the milk and cream and raise the heat a bit. Add the poblano puree and simmer to thicken and cook the sauce, 20-25 minutes.
Pre-heat the oven to 375°F.
Heat a tablespoon of EVOO in a large skillet over medium-high heat. Add the 1 remaining tablespoon butter and melt. Season the chicken breasts with salt and pepper and brown for 3-4 minutes on each side. Transfer to a casserole dish, spread the poblano sauce over the top and cover with cheese. Bake for 15-20 minutes, depending on the size of the chicken pieces – it should be brown and bubbly on top.
Serve garnished with cilantro sprigs and red onions, with charred corn tortillas or rice with green peas alongside.