Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a 10-inch springform pan.

Stir cake mix and instant pudding together in a large bowl; make a well in the center; add 1 1/4 cups water, eggs, and oil. Mix until well blended. Transfer to the prepared pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 45 to 50 minutes. Cool cake completely on a wire rack before removing from the pan.

Cut a 1-inch thick slice off top of cake and set aside. Scoop cake from the center, leaving 1-inch around border of cake. Discard or save for another use. Whisk milk and flour together in a saucepan over medium heat, stirring constantly until thickened. Set filling aside to cool completely.

Beat sugar, 1/2 cup shortening, butter, and vanilla together in a large bowl on high speed until light and fluffy. Add cooled milk mixture and continue to beat for 4 more minutes. Spoon filling into center of cake; replace reserved top layer.

Combine chocolate chips, remaining 1/4 cup shortening, corn syrup, and 2 1/4 tablespoons water in the top of a double boiler. Heat, stirring frequently, until melted and smooth. Pour warm glaze over cake and smooth over sides. Refrigerate until serving.