Preheat the oven to 350 degrees Fahrenheit. Line a 9 by 9 inch baking pan with parchment paper, leaving overhang on two opposite sides, and lightly coat the paper with nonstick spray.

Pulse the gingersnap cookies in a food processor until they become fine crumbs. In a medium bowl, stir together the crumbs, granulated sugar, and salt. Pour in the melted butter and mix until the mixture looks like wet sand. Press the crumb mixture firmly and evenly into the bottom of the prepared pan, then bake for 10 minutes. Set the pan aside to cool slightly.

In a large mixing bowl, beat the softened cream cheese with an electric mixer on medium speed until completely smooth, about 2 minutes. Add the granulated sugar and brown sugar and continue beating for 1 minute until light and fluffy. Add the eggs one at a time, mixing on low speed after each addition, then blend in the molasses and vanilla extract until smooth.

In a small bowl, whisk together the ground ginger, cinnamon, cloves, nutmeg, and flour. Sprinkle the spice mixture over the wet filling and mix on low speed just until no streaks remain.

Pour the cheesecake filling over the prebaked crust and spread it into an even layer. Tap the pan firmly on the counter a few times to release air bubbles. Bake at 350 degrees Fahrenheit for 35 to 40 minutes, until the edges are set and the center still has a slight jiggle. Let the pan cool completely on a wire rack, then refrigerate for at least 4 hours or overnight.

To make the frosting, beat the softened cream cheese and butter in a medium bowl until smooth and creamy, about 2 minutes. Add the powdered sugar and vanilla, beating on low speed until combined, then increase to medium speed for 1 minute until fluffy. Add the heavy cream or milk 1 tablespoon at a time until the frosting reaches a spreadable consistency.

Spread the cream cheese frosting evenly over the chilled cheesecake layer. Refrigerate the pan for 30 minutes to let the frosting set. Use the parchment overhang to lift the whole slab out of the pan, then slice into 16 squares, wiping the knife clean between cuts.