Preheat your oven to 350°F (175°C). Generously grease a standard bundt pan or a 9-inch round baking dish with butter.
Place the shredded mozzarella and cream cheese in a large microwave-safe bowl. Microwave in 30-second intervals, stirring between each, until fully melted and smooth. In a separate bowl, combine the almond flour and baking powder, then add the eggs and stir to form a crumbly mixture.Pour the melted cheese into the almond flour mixture and knead with your hands until a smooth, uniform dough forms.
In a small bowl, stir together the erythritol and cinnamon for the coating.
Pinch off small pieces of dough and roll them into 1-inch balls, about 25 to 30 pieces total. Roll each ball in the cinnamon coating until evenly covered, then layer them into the prepared bundt pan.
Bake for 22 to 28 minutes, until the dough balls are puffed, golden, and cooked through.
While the monkey bread bakes, prepare the caramel glaze. Melt the butter in a small saucepan over medium-low heat. Add the brown sugar erythritol and heavy cream, stirring constantly for 2 to 3 minutes until the mixture is smooth and slightly thickened. Remove from heat and stir in the vanilla extract.
Let the monkey bread cool in the pan for 5 minutes, then invert it onto a serving plate. Drizzle the warm caramel glaze over the top. Serve immediately and pull apart to eat.