Step 1

Heat the oil over medium heat in a large, heavy soup pot. When hot, add the potatoes and onions and cook, stirring often, until the onions are soft and translucent, about 4 to 5 minutes. Add the tomato, prepared Anaheim or poblano chiles, and salt, and cook until the ingredients are softened, 4 to 5 minutes more.

Step 2

Add the chicken broth, bring to a simmer and cook for 10 minutes, or until the potatoes are completely tender and the broth has thickened a bit. Taste and adjust salt. Reduce the heat to medium-low, slowly add the milk and bring back to a gentle simmer. Gradually crumble the cheese into the simmering soup and stir until cheese is completely melted – or serve in bowls with cubes of the cheese in the bowls adding the soup on top. Taste again for salt and serve hot.