Combine warm water, brown sugar, and yeast in a large mixing bowl and let rest until frothy, about 10 minutes. Add 1/2 cup oil and stir. Set aside.

Sift flour and salt together in a bowl.

Mix flour mixture into the yeast mixture and knead until dough forms a sticky ball. Coat the inside of another bowl with oil; transfer dough to the oiled bowl and turn to coat. Cover the bowl and let dough rest in a warm place until doubled in size, about 1 hour.

Meanwhile, mix white sugar and cinnamon together in a bowl.

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

Punch down risen dough and roll into 1-inch balls. Roll balls in cinnamon-sugar mixture and arrange a layer in the prepared tube pan. Sprinkle with cinnamon sugar and drizzle with a bit of oil. Continue layering with remaining dough balls and cinnamon sugar.

Bake in the preheated oven until puffed up, about 25 minutes.

Meanwhile, mix confectioners' sugar and milk together in a bowl. Drizzle glaze generously over the warm monkey bread.