Preheat the oven to 325°F.
Grease the bundt pan: Dip a paper towel in a generous amount of shortening and spread the shortening around the pan, making sure to coat all of its nooks and crannies. Lightly dust the greased pan with flour (using a sift works well for this). Tap and rotate the pan to create a thin, even coating of flour across the pan’s interior. Gently tap out any excess flour over the sink.
Combine the cake flour, pudding mix, baking powder and salt together in a large bowl. Set aside.
In the bowl of a stand mixer fitted with a paddle attachement, cream the softened butter and sugar on medium speed for ~3-5 minutes or until butter becomes noticeably lighter in color and fluffy. Scrape down sides of bowl as needed.
Meanwhile, whisk together the sour cream and milk in a measuring cup or small bowl.
Decrease mixture to low and beat in the eggs, one at a time. Beat each egg just long enough for the yellow to disappear (about 30 seconds). Then add in vanilla extract.
Gradually add the flour mixture alternately with the sour cream mixture, beginning and ending with the flour mixture. Beat just until combined after each addition. Over mixing can lead to dry, tough cakes.
Transfer half of the batter to the prepared bundt pan. Top with 1/2 cup of the flour coated chocolate chips, pour remaining batter on top and smooth it out. Top with the rest of the chocolate chips and swirl chips into the cake with a toothpick. Gently tap pan to release any air bubbles.
Bake until a skewer inserted into the cake comes out clean or with only a few moist crumbs, ~45-50 minutes. Be careful not to over bake. Cake will continue to cook in pan after you pull it out of the oven.
Cool cake in pan for 10 minutes. Cooling allows cake to slowly begin to release itself from the sides of the pan. Resist the urge to pull it out immediately.
Make the simple syrup while the cake cools: Pour sugar into a small saucepan and then immediately add water. Bring to a boil and stir until all sugar has dissolved. (1-2 minutes). Remove from heat and stir in vanilla. Set aside.
Invert cake: Place a parchment paper lined rack or baking sheet over the pan, then flip both over in one motion. Slowly lift the pan from the cake.
Poke holes all around the cake with a fork or toothpick and brush the syrup over the cake using a pastry brush, letting it soak into the holes. Allow the syrup to sink in to the cake, then add more syrup. Repeat until all syrup is used.
Allow cake to cool, then make the vanilla bean frosting. Whisk ingredients for glaze together into a medium sized bowl and pour glaze over the cake. Garnish with more mini chocolate chips if desired. Let sit until set before slicing the cake and serving. Enjoy!