In large soup pot, cover chicken and poaching ingredients with about 2 1/2 to 3 quarts water. Bring to a boil then reduce heat to low. Simmer and poach 1 hour. Remove chicken breasts to a plate to cool. Peel and remove bones from chicken and slice the breast meat. Strain the poaching liquid and reserve.
Preheat broiler. Place poblano and red bell peppers on a baking sheet and char evenly to blacken skins all over – leave oven door slightly ajar to allow the steam to escape. Place peppers in a large bowl and cover with plastic wrap to steam peppers and loosen flesh from skin.
In a small pot, cover potatoes with water and bring to a boil. Salt water and cook potatoes until very soft. Drain and mash until very smooth with some of the reserved stock.