Combine the rice and water in a heavy 3-quart saucepan. Bring to a boil, reduce to low, cover, and cook for 10–12 minutes, until most water is absorbed. The rice does not need to be fully tender yet.

Stir in the milk and salt. Bring to a gentle simmer over medium-low heat, then reduce the heat as needed. Cook uncovered for 25–35 minutes, stirring frequently and scraping the bottom and corners, until the grains are tender without a chalky center. Avoid a rolling boil. If the rice is still firm and the mixture looks dry, add water a tablespoon at a time and continue cooking.

Stir in the pumpkin, sugar, cinnamon, and ginger. Simmer gently for 5–10 minutes, stirring often, until the sauce coats the rice but remains slightly loose. The timing includes bringing the pot back to a simmer; cook time is approximately 65 minutes overall and may vary with the pan.

Remove from the heat and stir in the vanilla. Rest for 10 minutes, stir, and divide into 6 servings. It thickens further as it cools.

For leftovers or cold servings, transfer promptly to shallow containers, cool quickly, ideally within 1 hour, and refrigerate at 40°F or below. Use within 24 hours. Reheat a portion only once, stirring until steaming hot throughout; never leave the whole saucepan on the counter to cool.