Measure out 1/2 cup of the cake mix and set it aside.
Put the rest of the cake mix, the melted butter, and 1 egg in a large mixing bowl. Blend together with a mixer for 2 minutes, scraping down the sides of the bowl occasionally. The batter should come together into a ball. With your fingertips, pat the batter evenly over the bottom and 1 inch up the sides of the pan.
Place the cream cheese and condensed milk in the same mixing bowl you used for the crust and blen together with a mixer until just combined - about 30 seconds. Stop the mixer and add the reserved cake mix, the remaining 3 eggs, the pumpkin, brown sugar, cinnamon, and ginger. Beat on medium speed for 1 minute, scraping the sides of the bowl occasionally. Pour the filling onto the crust and spread it with a spatula so it covers the entire surface. Place in oven.
Bake the cheesecake until it looks shiny and the center no longer jiggles when you shake the pan - about 45 to 49 minutes. Remove the pan from the oven while you prepare the topping, but leave the oven on.
For the topping, place the sour cream and brown sugar in a small bowl and stir with a spoon until well combined. Spread over the cheesecake.
Bake until the topping sets - about 7 to 8 minutes. Remove pan from oven and place on wire rack for 30 minutes. Then lightly cover the pan with plastic wrap and refrigerate it for at least 1 hour or preferably 24 hours.