Preheat the oven to 350 degrees F (175 degrees C). Grease a fluted tube pan (such as Bundt®).

Mix together sugar and cinnamon in a medium bowl. Tear each biscuit in 1/2, then roll each piece into a ball. Roll each ball in the sugar-cinnamon mixture until completely covered. Place in the prepared pan, filling the pan up to 3/4 full to allow biscuit balls to expand.

Combine butter, vanilla extract, and nutmeg in a small bowl, mix in some of the leftover sugar-cinnamon mixture, and pour over the biscuit balls.

Bake in the preheated oven until golden brown and no longer doughy in the center, about 45 minutes. Cool in the pan for 5 to 10 minutes, then flip pan onto a serving dish to release the monkey bread.