arm the skyr/Greek yogurt in a microwave-safe bowl for 30 seconds until slightly warm to the touch. Set aside.
Proof the yeast by warming the milk to 110 degrees F. Combine the warm milk, yeast, and a small pinch of the measured sugar in a small bowl. Stir gently and let sit for 5 to 10 minutes until foamy and fragrant. If the yeast does not foam, it is not active and should not be used.
Combine the bread flour and remaining sugar in the bowl of a stand mixer fitted with the dough hook. Whisk briefly to distribute.
Add the foamy yeast mixture, warmed skyr/Greek yogurt, egg plus yolk, and vanilla to the flour. Mix on low speed until a shaggy dough forms, about 2 minutes.
Add the salt and softened butter one tablespoon at a time with the mixer running on low speed.
Increase the mixer speed to medium and knead for 8 to 10 minutes until the dough is smooth, supple, and pulls cleanly from the sides and bottom of the bowl. The dough will feel slightly tacky, which is normal.
Let the dough rest uncovered for 10 minutes after kneading. Do not add extra flour during this rest.
Shape the dough into a ball and place it in a lightly oiled bowl. Cover and let rise in a warm spot at 75 to 80 degrees F until doubled, about 60 to 90 minutes.
While the dough rises, whisk together both sugars, the cinnamon, and cornstarch in a 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 risen 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. Want your balls larger or smaller adjust your grid. Its totally customizable.
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 let rise in a warm spot for 45 to 60+ minutes until the balls are puffed and touching. Your kitchen temperature will dictate the speed of your proof.
Preheat the oven to 350 degrees F during the last 15 minutes of rising.
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.