Preheat oven to 350°F (175°C) and line an 8 × 8-inch pan with parchment.
Whisk flour, ginger, cinnamon, cloves, baking soda, and salt.
Beat butter and brown sugar until creamy.
Mix in molasses, egg, and vanilla.
Add dry ingredients and mix just until a soft dough forms.
Press the dough evenly into the prepared pan.
Bake the gingerbread base for 10 minutes.
Beat the cream cheese until smooth.
Mix in granulated sugar.
Add eggs one at a time, mixing gently.
Mix in vanilla and sour cream.
Spread cheesecake batter over the partially baked gingerbread base.
Bake for another 28–35 minutes, until the edges are set and the center has a slight wobble.
Cool completely, then refrigerate for at least 4 hours.
Beat cream cheese and butter for the frosting until smooth.
Add powdered sugar, vanilla, and salt and mix until creamy.
Spread a thin layer over the chilled cheesecake.
Lift from the pan and cut into 16 bars.