Dice your poblano pepper and onion. Peel your garlic cloves and press the cloves (or finely chop them). Take out your cream cheese and cube it. Open your cans of diced green chilis and great northern beans. Drain the liquid from the green chilis, but not the beans.
In a large pot, heat a large drizzle of olive oil. Add in the poblano pepper, onion, and green chilis. Season with some salt and pepper to taste. Sauté on medium-high heat until the onion starts to become translucent. About 5-8 minutes.
Add in the garlic and saute for an additional 1-2 minutes.
Add the cumin, chili powder, oregano, and a touch more salt and pepper if needed. Sauté the spices together with the peppers and onions until everything in well incorporated. This will take another 1-2 minutes.
Taking the chicken broth from your cooked chicken, top it off with a little water to assure there are 2 cups of broth. Pour this into the pot with the peppers, onion, and seasonings. Allow this to heat for a couple of minutes.
Add in the cubed cream cheese and heat on medium-high while stirring the cream cheese in. If there are clumps that are not stirring in, try to break them up with your spoon until everything is pretty well incorporated. There can be some small pieces, this will all mix in as it cooks.
Add in the pepper jack cheese and colby jack cheese. Allow this to simmer and heat until all the cheese is incorporated into the liquid.
Pour in the heavy cream and stir to combine. Then add the sour cream, great northern beans and salsa. Stir all ingredients to combine. Allow everything to come to a boil, still over medium-high heat.
Once the mixture has come to a boil, reduce the heat to medium-low, allowing the chili to simmer. Add in the shredded chicken and stir to combine.
Allow all ingredients to cook on medium-low for at least 30 minutes. Stir this frequently, as a thicker chili like this can start to burn if not stirred.
Serve immediately with the desired toppings. See my list of suggested topping ideas in the post above. I highly suggest at least adding some fresh lime juice.