Grease a standard bundt pan with softened butter.
In a large mixing bowl with the paddle attachment add gluten-free flour, baking powder, instant yeast, salt and sugar. Mix to combine.
Next, while the mixer is running on low, add the milk, eggs and softened butter.
Turn the speed up to medium and mix for 3 minutes until very smooth. The dough will be thick but that is normal. Take an oiled spatula and scrape down the bottom and sides of the bowl. Cover the bowl tightly with plastic wrap and let rise for 45-60 minutes, until doubled in size.
While the dough is rising, prepare the coating.
In a microwave-safe bowl, melt the butter.
In another more shallow bowl, mix the white sugar, brown sugar, and cinnamon.
Once the dough has risen, pull small dough chunks and roll into 1-inch balls.
Dip each ball, one by one, in the melted butter and then generously roll in the cinnamon-sugar mixture to coat them well. You may not use all of the cinnamon-sugar or you may need more cinnamon-sugar depending how heavy you coat each ball.
Arrange the balls in the bundt pan as you go. Cover the pan with aluminum foil, plastic wrap, or a clean kitchen towel and allow the shaped monkey bread to rest for 20 minutes. The balls will slightly rise during this time.
Adjust oven rack to a lower position and preheat oven to 350°F (177°C).