First, make the banana pudding bundt cake. Position a rack in the center of the oven and preheat the oven to 350°F. Line a half sheet pan with parchment paper.
Mix the dry ingredients. In a medium bowl, whisk together the flour, instant banana pudding mix, baking powder, and salt.
Mix the wet ingredients. In another medium bowl, whisk together the buttermilk, oil, and banana extract.
Whisk the eggs and sugar. In the bowl of a stand mixer fitted with the whisk attachment, whisk the eggs. Whisk on low speed for 1 minute, then add the sugar.Turn the mixer up to medium-high and whisk for 5 minutes, or until thick and doubled in volume. Mix the wet ingredients, followed by the dry ingredients, into the egg mixture. Use a rubber spatula to scrape down the bottom and sides of the bowl, and swap the whisk attachment with the paddle attachment.With the mixer on low, slowly pour in the wet ingredients, and beat until just combined.Scrape down the bottom and sides of the bowl, then gradually add the dry ingredients and beat until just combined. Prep and fill the bundt pan. Generously spray a 12-cup capacity bundt pan with nonstick baking spray. Immediately pour the batter into the prepared bundt pan. Set the filled bundt pan on the sheet pan. Bake the bundt cake. Bake for 65 to 75 minutes, or until a skewer inserted into the cake comes out with a few crumbs attached. Cool on a wire rack for 10 minutes. Cool the bundt cake. After 10 minutes, place a serving plate or second wire rack over the bundt cake and flip the cake to turn it out of its pan. Cool completely before glazing.
Make the vanilla glaze. Once the cake is cool, make the honey glaze. In a medium bowl, whisk together the confectioners’ sugar, milk, vanilla, and salt for the glaze until smooth.
Glaze the cake. Pour the glaze over the top of the cake and use an offset spatula to spread it into a thin layer on top of the cake. Let any excess glaze drip off and transfer to a wire rack to dry for 1 to 2 hours.
Serve and store. Serve at room temperature. The assembled banana bundt cake can be stored at room temperature, under a cake dome or a large bowl turned upside down, for up to 3 days. Press a sheet of plastic wrap against any cut surfaces to prevent the cake from drying out.