Warm the skyr/Greek yogurt and milk together in a microwave safe bowl for 30 to 45 seconds until slightly warm but not hot. The mixture should feel comfortable to the touch. If it is too hot it can affect your starter.
Whisk together the bread flour, baking powder, salt, and sugar in a large bowl.
In a separate bowl whisk together the starter, warmed skyr/Greek yogurt and milk mixture, vanilla, and the egg plus yolk until smooth. Mix in the softened butter.
Pour the wet mixture into the dry ingredients and mix until a soft, slightly tacky dough forms.
Knead the dough in a stand mixer with the dough hook for 5 to 6 minutes until smooth and elastic, aiming for a partial windowpane. The dough will feel softer and more relaxed than a lean sourdough, which is correct for an enriched dough.
Place the dough in a lightly oiled bowl and cover. Perform one set of stretch and folds after the first 30 to 45 minutes, then leave it alone.
Bulk ferment at room temperature for 6 to 10 hours until the dough has increased 50 to 75 percent in volume and looks domed and slightly jiggly. At cooler temperatures around 65 degrees F this can take up to 12 hours overnight.
While the dough finishes fermenting, whisk together both sugars, the cinnamon, and cornstarch in a shallow bowl until fully combined. Melt the butter for the vanilla butter in a separate bowl and stir in the vanilla. Line a 9 by 13 inch pan with parchment paper and set aside.
Turn the fermented dough out onto a lightly floured surface and press into a rough 10 by 10 inch square. Use a bench scraper to cut a 5 by 6 grid, giving you 30 equal pieces at approximately 23 grams each. Roll each piece briefly in your palms into a loose ball.
Pour the vanilla butter into the parchment lined pan and spread it evenly across the bottom. Add the dough balls and give them one quick flip so every piece is coated on all sides. Sprinkle half the cinnamon sugar mixture evenly over the top. Flip each ball individually, then sprinkle the remaining cinnamon sugar over the top of the balls, keeping as much of the coating on the surface of each ball as possible rather than letting it fall between them.
Cover the pan loosely and proof at room temperature for 3 to 4 hours until the balls are puffed, touching, and pass the poke test. Press gently with one finger and the dough should spring back slowly, leaving just the faintest impression. Preheat the oven to 350 degrees F during the last 20 minutes of proofing.
Bake for 25 to 30 minutes until deep golden brown and the internal temperature of a center ball reads at least 190 degrees F on an instant read thermometer. If the tops are browning too quickly, tent loosely with foil at the 20 minute mark.
Let the pan cool for 5 to 10 minutes. While it cools, whisk together the powdered sugar, vanilla, salt, and 2 tablespoons of milk or cream until smooth and pourable. Add the third tablespoon if needed to reach a drizzleable consistency.
Drizzle the glaze over the warm monkey bread and serve immediately, pulling pieces apart with your fingers.