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Preheat oven to 375 degrees F. Prepare water for pasta and once it comes to a boil, cook pasta according to directions, then drain.
While water is boiling, heat a large cast-iron (or oven-safe) skillet over medium-low heat. Add olive oil, then toss in squash with salt, pepper and nutmeg. Add stock and cover skillet, cooking for 15-20 minutes, stirring occasionally, until squash is soft and can easily be mashed. Remove cover and reduce heat to low. Mash squash with a potato masher or forks, smoothing it out with a spoon to remove all of the chunks. Add in milk, 4 ounces of fontina and all of the parmesan cheese, stirring until melted and smooth, about 5 minutes. If mixture still seems way too starchy, add in additional milk 1/4 cup at a time, stirring for a while. Stir in brown butter, again mixing for a minute or so until incorporated. Taste and season additionally if desired.
Fold shells into the sauce, taking a few minutes to fully toss the shells so they are all coated. Top with remaining fontina cheese (or as much as you see fit) and breadcrumbs and bake for 15 minutes. If a bubbly, golden topping is desired, broil for 1-2 minutes on high after cooking. Sprinkle with fresh herbs and serve!