Preheat oven to 400 degrees F (200 degrees C). Place butternut squash halves cut-side down on a baking sheet.
Roast in the preheated oven until tender when pierced with a fork, 45 to 60 minutes. Let cool, about 15 minutes. Scoop flesh into a large bowl and mash until smooth.
Mix flour, oats, cinnamon, ginger, and nutmeg together in a bowl.
Beat butter and shortening in a bowl until blended, about 30 seconds. Add white sugar and brown sugar; beat until fluffy. Add mashed squash, egg, milk, and vanilla extract; beat thoroughly until dough is combined. Fold in flour mixture.
Chill dough until firm, about 2 hours.
Preheat oven to 375 degrees F (190 degrees C). Line 2 baking sheets with parchment paper.
Drop 1-inch scoops of dough onto the baking sheets.
Bake in the preheated oven until edges are golden brown, 10 to 12 minutes. Transfer to a wire rack to cool before serving.