If using dried navy beans, soak beans overnight in enough cold water to cover them by 2 inches. Drain beans in a colander and return to kettle with cold water to cover by 2 inches. Cook beans at a bare simmer until tender, about 1 hour, and drain in colander. If using canned navy beans, rinse and drain the beans well and set aside.
In a large heavy-bottom soup pot, heat 2 Tbsp. butter over medium-high heat. When melted, add the chopped chicken and cook until the outside of the chicken pieces are no longer pink (don't overcook the chicken - it will finish in the soup.) Using a slotted spoon, remove the chicken from the pot and set aside.
Reduce the heat to medium and add the remaining 2 Tbsp. butter and onion. Cook until the onion becomes soft, about 3 minutes. Add the flour, chili powder, and cumin. Cook, stirring constantly until the flour begins to brown and stick to the bottom of the pan, about 3 minutes. Gradually add chicken broth, whisking constantly. Bring mixture to a boil and simmer until it begins to thicken, about 5 minutes.
Add the navy beans, prepared chicken, salsa verde, and sweet corn, and simmer for 15 minutes. Remove from heat and stir in the Monterey Jack cheese and greek yogurt. Add salt and pepper to taste. Squeeze the juice from the lime wedges directly into the soup and stir to combine.
Serve immediately with cheese, sour cream, salsa, and cilantro on the side.