Preheat the oven to 350°F (180°C). Grease 12-cup fluted bundt pan with nonstick spray.
In a small bowl, mix together the sugar and cinnamon.
Open the cans of cinnamon rolls. Cut each roll in half, then flatten each piece into a circle with your fingers. Place a piece of cream cheese in the center of the dough round. Fold the dough over the cream cheese to seal and roll into a ball. Toss in the cinnamon sugar. Place in the prepared bundt pan. Repeat with remaining dough pieces.
In a small bowl, whisk together the brown sugar and melted butter until smooth. Pour the mixture over the filled cinnamon roll balls.
Bake the monkey bread for 35–40 minutes, or until golden brown. Let cool in the pan for 5–10 minutes, until cool to the touch, then invert onto a serving platter.
Add the frosting from the cans of cinnamon rolls into a small microwave-safe bowl. Warm in the microwave for 30 seconds to loosen.
Drizzle the icing over the monkey bread. Serve warm.