To dry out the bread, cut it into cubes and heat on a baking sheet in a 350 degree oven for 10-15 minutes. Or cut it into cubes and let it sit out on the counter uncovered for 24 hours.
Preheat the oven to 400 degrees.
Toss butternut squash with 1 tablespoon of olive oil and ½ teaspoon salt and pepper. Bake on a parchment-lined baking sheet for 15 minutes.
Meanwhile, cook the sausage in a saucepan over medium-high heat for 6-8 minutes. Add garlic, onion, celery, sage, and thyme. Sauté for 3 more minutes then remove from heat.
Remove squash from the oven after 15 minutes, then turn the oven down to 375.
Beat the eggs, and stir them into the chicken stock.
In a large bowl combine the bread cubes, sausage mixture, baked squash, Parmesan and Asiago cheese, and ½ teaspoon salt and pepper.
Pour half the chicken stock/egg mixture into the bowl, distributing evenly, and gently fold together. *When mixing the bread with the other ingredients, avoid over-work the stuffing or the bread will lose its structure.
Pour the rest of the chicken stock liquid and the melted butter evenly over the mixture, and gently fold together again.
Spray a 9" X 13" dish with cooking spray and add the stuffing mixture.