Place the oven on Broil to broil and roast the peppers.
Broil the peppers for 5-10 minutes (cook time will vary based on the size of your peppers) or until the skin of the peppers darken and form blisters.You can also air fry the peppers on 400 degrees for 5-10 minutes (cook time will vary based on the size of your peppers).
Remove the peppers from the oven and place them in a paper bag or plastic zip lock bag. I prefer paper bag. Allow the peppers to cool. This will steam the peppers and loosen up the skin. While the peppers cool I prep the remaining ingredients.
Once the peppers have cooled, remove the skin from the outside of the peppers and chop the peppers into smaller chunks, a couple of inches in size.
Add the butter or olive oil to a Dutch oven or soup pot on medium-high heat. Once melted, add chopped onions, carrots, celery, and garlic. Cook for 3-4 minutes until the onions are translucent and fragrant.
Reduce the heat on the stove to medium. Add the broth and chopped peppers. Stir to combine the ingredients.
Use an immersion blender or pour the soup mixture into an actual blender. Pulse the ingredients until the soup is smooth. Be careful not to splatter the soup and burn yourself.
If using a standard blender, return the soup to the pot. Add in the milk (or half and half), shredded cheese, optional lime juice, salt, and pepper to taste.
Continue to cook on Low and stir until the cheese has melted and the soup has thickened.