Boil cinnamon stick and 2 cups water in a heavy medium saucepan over high heat. Stir in rice. Cover and reduce heat to low. Cook rice 20 minutes or until tender. When done, leave pan on heat, remove cinnamon stick and discard it.
Add the milk, 1/2 cup sugar and salt to the cooked rice. Mix well and increase heat to medium. Bring to a low boil. Cook, stirring frequently, approximately 10 minutes or until pudding is very thick (a spoonful should hold its shape firmly when scooped up). Remove from heat and set aside. Cool completely.
In a small bowl, combine remaining 1/4 cup sugar and 1 teaspoons ground cinnamon for topping. Set aside.
Measure about 1/2 cup rice pudding and place in center of each tortilla. Carefully fold each filled tortilla in half, pressing gently to seal the edges (you should not need water or egg for this, the rice pudding should be sufficient to keep them closed).
Pour oil in a large cast iron skillet (approx. 1/8" in depth). Heat over medium heat until oil is hot but not smoking (oil will appear to shimmer when it it hot enough). Fry empanadas in batches until crisp and golden brown (approximately 2 to 3 minutes per side). Remove from oil and set on paper towels to drain excess oil. Sprinkle reserved cinnamon sugar topping generously on both sides of warm empanadas and serve immediately for best flavor.