For the pumpkin cheesecake filling: Make the filling ahead of time and chill while you are making your cake. Add the cream cheese to the bowl of a stand mixer fitted with the paddle attachment and beat on medium-high speed until light and fluffy, 2 to 3 minutes. Add the pumpkin, vanilla and pumpkin pie spice and beat on medium speed just until incorporated. Change the mixer speed to low and add the powdered sugar 1/2 cup at a time until all the sugar is incorporated, then increase the mixer speed and beat for 2 more minutes. Fold in the pecans. Chill while you bake the cake. (See Cook's Note.)
For the cake: Preheat the oven to 450 degrees F. Spray a 16-by-11-inch jelly roll/sheet pan with nonstick baking spray. Line the pan with parchment paper and spray again lightly with baking spray. Set aside.
Whisk together the cake flour, cornstarch, ginger and salt in a large bowl.
Put the eggs, egg yolks and 1/2 cup granulated sugar in the bowl of a stand mixer fitted with the paddle attachment. Beat on medium-high speed until pale in color and thickened, around 5 minutes. Add the vanilla extract and beat a few more seconds to combine.
Sift the flour mixture over the egg and sugar mixture and gently fold with a spatula until combined.
Add the egg whites and remaining tablespoon granulated sugar to a separate grease-free bowl and whip until stiff peaks form.
Gently fold the whipped egg whites into the egg yolk mixture with a spatula. Remove 1/4 cup batter to each of 3 separate bowls and add a couple drops of orange food color to one, red to another and brown to the last. Stir just to incorporate.
Brush an abstract pattern with the colorful batter onto the parchment. Freeze at least 10 minutes, and up to 30 minutes. Pour the remaining plain batter over the top of the design and smooth gently to even out. Bake until lightly golden brown and the cake pulls away from the edges, 6 to 7 minutes.
Let the cake sit at room temperature for 5 minutes. Sift the entire surface with powdered sugar. Cover with a clean towel and use the bottom of a sheet pan to flip the cake out of pan onto the towel. Remove the pan and parchment paper. Cover the exposed top of the cake with a second clean kitchen towel and again use the bottom of a sheet pan to flip the cake back over. Remove the towel from the top of the cake and use the towel that is underneath the cake to help roll it into a tight roll, rolling short end to short end. Set aside to cool completely.
Gently unroll the cake and spread the filling in an even layer over it. Roll the cake back up using the towel as a guide. Place on a sheet pan or your serving platter and refrigerate for 2 hours before slicing and serving.
I recommend making this filling ahead and chilling overnight; the longer it chills, the easier it will be to work with.