Preheat oven to 350°F. In an oven-safe dish, sit squash and place butter and salt inside. Bake in preheated oven, 30 minutes. Remove and cool for about 5 minutes. Carefully scoop meat out of the squash leaving a ½” thick rim inside.
In a saucepan over medium heat, combine squash meat and cider, about 10 minutes. Stir in heavy cream and cream cheese. Stir constantly until melted. Add shredded cheese, nutmeg, and thyme. Continue blending over a medium heat until the fondue is fully blended (for a smoother consistency, use an immersion blender; otherwise, work to break up the squash). Pour cheese mixture into the acorn squash shell.
Serve immediately with bread, cherry tomatoes or anything else you love fondue on.