Into a medium-sized soup pot over medium-high heat, add the butter. When the butter is melted, add the onion. Cook for 2 minutes, stirring frequently until softened. Add the garlic and continue cooking for an additional 30 seconds.

Whisk in the flour, then add the potatoes and broth. Bring just to a boil then reduce the heat to low. Cover and simmer for 15 minutes, or until the potatoes are fork-tender. Stir in the milk, chiles, salt, and pepper. Continue cooking without boiling for 3 minutes, until heated through.

Add the cheddar cheese, stirring occasionally until melted. Adjust seasoning to taste and serve hot.