Preheat oven to 350. Cut each squash in half length wise and remove seeds. Place squash flesh side down on a greased cookie sheet. Bake for 45-60 minutes or until skin side is brown and bubbly and flesh side is golden brown.
While squash is roasting, peel and cube onion, chop celery, peel and halve garlic cloves, and peel and chop carrots. (It doesn't really matter how big you make everything, because it gets blended later. The only thing to keep in mind is the bigger the pieces, the longer it takes to get cook.).
Melt the butter in a large stock pot. Add the onion, celery, and garlic. Cook until everything is beginning to get soft and has a little color. Add stock and bring to a boil. Add carrots. Reduce heat and simmer until carrots begin to soften.
At this point, you have 2 options. The chicken can be blended along with everything else (sounds a little weird, but I promise it is delicious), or you can add the chicken after blending. If you are going to blend the chicken: Cube it (again size isn't a huge issue for this option) and add it to the simmering pot. Simmer until the chicken is cooked through and the carrots are soft.
Take squash out of oven and let cool until you can handle it. Using a spoon, scoop the cooked flesh out of the skin. Add it to the simmering pot. Simmer until the squash has broken down. Remove from heat and let cool slightly.
In small batches, blend soup until smooth. (Because you are blending something hot, you will want to take the center out of the lid of your blender to vent it. Cover the hole by holding a folded up kitchen towel over it.) Once you've got all of it blended, put the blended soup back in the pot.
If you didn't blend the chicken, this is where you would cube it small and add it in, simmering until it's cooked through.
Add salt and pepper to taste. At this point, I add brown sugar, but you don't have to. Also, you can add whipping cream, but I find it's creamy enough without it and I don't miss it.