Set a large sauce pot over medium heat. Add the butter, onions, and garlic, and sauté for 3-5 minutes. Scoop the softened onions into the food processor (or blender) and add one cup chicken broth. Puree the onions and broth until completely smooth.
Pour the onion mixture back into the sauce pot, and add the remaining broth and both bottles of beer. Bring to a boil, then lower the heat.
Toss the shredded Irish cheese with cornstarch. Add the shredded cheese, cream cheese, and Dijon mustard to the broth. Whisk to melt the cheese and break up the cream cheese cubes. Whisk until the soup is completely smooth and slightly thickened, about 10 minutes. (Do not boil the cheese! Lower the heat if needed.)
Taste, then salt and pepper as needed. Serve as-is, or garnish with bacon and chives if desired.
The beer broth will foam up quite a bit when boiling. Make sure to use a large pot so it doesn't boil over.