In a large, heavy pot over medium heat, cook milk, ice cream, sugar, cinnamon, and salt, stirring frequently, until ice cream melts and sugar is dissolved, 5 to 6 minutes.
Stir in rice; bring to a gentle boil. Immediately reduce heat to medium-low, bring to a simmer, and cook, stirring occasionally to avoid scorching and adding vanilla during the last 10 minutes of cooking, until desired consistency is reached, 45 to 50 minutes. Note: While the pudding is hot, this will be the loosest it will be, so look for a texture that is slightly looser than what you want, and it will reach the desired thickness when cooled.
Let cool to room temperature. Transfer to an airtight container and serve at your desired temperature; it tastes best after at least 1 night in the refrigerator. Top with desired toppings before serving.
