Roast Poblano chiles over a gas flame, turning frequently with tongs, until charred on all sides, about 5 minutes each. Place in a plastic bag, set aside to allow steaming for 10 minutes.
Then, using your hands, rub and remove the pepper-charred skin under running water (you may want to wear latex gloves or plastic bags on your hands as you work with the chilies, to avoid inadvertently touching your eyes later with pepper-coated fingers). Cut chiles open, remove, stem, seeds and veins. Chop in strips and reserve. Save a few for garnish.
In a large, heavy saucepan, heat the butter and oil over medium heat. Stirring frequently, cook onion and garlic until softened. Add potatoes, water, chicken stock, corn and parsley. Bring to boil, then reduce heat to simmer. Cook until potatoes are very tender, about 15 minutes.
When potatoes have softened, transfer to a blender or food processor. Add chile strips, cream and salt. Puree until smooth. Return soup to the saucepan and reheat to serving temperature, stirring.
Serve garnished with Poblano chile strips and cubed cheese.