Preheat the oven to 350°F and line a 9x9-inch baking pan with parchment, leaving an overhang on two sides. Mix the gingersnap crumbs with the melted butter, press firmly into the pan and bake for 10 minutes.
Reduce the oven to 325°F and place a pan of hot water on the lower rack. Blend the cottage cheese in a food processor or blender for 2 to 3 minutes, scraping down the sides, until completely silky.
Add the cream cheese, sugar, cornstarch, vanilla and salt and blend until smooth. Add the eggs and pulse just until combined.
Transfer 1 cup of the cheesecake batter to a separate bowl and whisk in the pumpkin puree, brown sugar, pumpkin pie spice and egg yolk until smooth.
Pour the plain batter over the crust and spread it evenly. Drop spoonfuls of the pumpkin batter over the top, then drag a skewer through both layers in a figure-eight pattern to create a marbled swirl.
Bake on the middle rack for 40 to 45 minutes, until the edges are set and the center has a slight wobble.
Let the bars cool completely in the pan, then refrigerate for at least 4 hours. Lift out using the parchment and cut into 16 bars.