Preheat your oven to 325°F (165°C). Line a 9-inch square baking pan with parchment paper, leaving overhangs on two sides to lift the bars out later.
In a medium bowl, combine the gingersnap crumbs, 3 tablespoons of sugar, and ½ teaspoon of cinnamon. Pour in the melted butter and stir until the crumbs are evenly moistened. Press the mixture firmly into the bottom of the prepared pan. Bake for 13-15 minutes, then set on a wire rack to cool slightly.
In a large bowl, use an electric mixer on low speed to beat the room temperature cream cheese and ¾ cup of sugar until smooth. Scrape down the bowl, then add the sour cream, vanilla, 1 teaspoon of ginger, 1 teaspoon of cinnamon, nutmeg, and cloves. Mix until just combined.
Add the eggs one at a time, mixing on low speed after each one just until the yolk is incorporated. Be careful not to overmix.
Transfer ½ cup of the batter to a small bowl. Stir the molasses into this smaller portion of batter until uniform. The color will darken.
Pour the plain cheesecake batter over the cooled crust and spread it evenly. Drop spoonfuls of the molasses-spiced batter over the top.
Use a skewer or thin knife to gently swirl the darker batter into the lighter batter. Make shallow swirls on the surface without touching the crust below.
Bake for 36-40 minutes. The edges should be set, but the center will still have a slight jiggle. For safety, the center should reach an internal temperature of 150°F (65°C).
Turn off the oven, prop the door open slightly with a wooden spoon, and let the cheesecake cool inside for 30 minutes. Then, remove it from the oven and let it cool completely at room temperature. Cover and refrigerate for at least 6 hours, or overnight, before serving.