Preheat the oven to 350ºF. Generously butter a 12 cup bundt pan.
In a large bowl, whisk together all-purpose flour, whole wheat flour, baking soda, and salt.
In a stand mixer fitted with the paddle attachment, beat the butter until creamy. Add the sugar and beat on medium speed until light and fluffy. Beat in the vanilla, then add the eggs one at a time, beating for about 1 minute after each addition. Reduce the mixer speed to low and add the bananas. Finally, mix in half the dry ingredients, then all of the yogurt, then the remaining dry ingredients. Transfer the batter to the prepared pan and rap it on the counter a few times to remove any air bubbles and smooth the top.
Bake 65-75 minutes, or until a thin knife inserted deep into the center comes out clean. If the cake is browning too quickly after 30 minutes, cover it loosely with a foil tent. Transfer the cake to a rack to cool for 10 minutes before unfolding onto the rack to cool completely.
Once the cake is cool, assemble the glaze by mixing powdered sugar and vanilla extract in a small bowl with a fork. Add milk a few drops at a time until the desired consistency is reached. It should be smooth enough to drizzle over the cake but not so thin that it becomes translucent. Drizzle evenly over the cake and allow to dry before covering with plastic wrap and storing in the refrigerator for up to 5 days.