Preheat the oven to 425 degrees.
Half the squash, lengthwise, and scoop out the seeds with a spoon. Place the squash, skin-side down, on a foil-lined baking sheet. Drizzle with olive oil and roast, uncovered, for 1 hour.
In a large soup pot, drizzle olive oil and add onion and celery over medium heat. Add a pinch of salt and pepper and then add garlic until fragrant.
Add beer and chicken stock and bring to a boil. Immediately reduce heat to a low simmer and cover.
In another medium pot, melt butter over medium-low heat. Add flour and whisk. Slowly add milk and cream while continuously whisking to avoid lumps. (Don't worry too much about the consistency because you will be pureeing the soup later.)
Add 3/4 of the cheese and continue to whisk, until melted into the sauce. Season with salt and pepper and remove from heat.
Be sure the soup is simmering, but not boiling. Using a ladle, add cheese sauce one ladle at a time, stirring continuously.
At this point, the squash should be fork tender. Remove from oven and allow to cool enough to be handled. Using s spoon, scoop out the squash from the skin and add to the soup.
Using a hand blender/wand, puree the soup in the soup pot to desired consistency. Add salt and pepper to taste.
To serve, garnish with chives and remaining cheese.