In a large saucepan, bring 3 cups water plus the salt to a boil. Add rice and stir. Reduce heat, cover and simmer on low heat for 20 minutes. Remove from heat, fluff rice with fork, re-cover, and let sit for 5 min.
Add to the rice: 1 can evaporated milk, 1.5 c hot water, and sugar. Over medium heat, stir thoroughly and heat until heated throughout; continue cooking until thick and creamy, about 15 minutes, stirring frequently.
Meanwhile, beat the eggs and leave at room temperature while rice is cooking. After the 15 min of cooking in the step above, warm the remaining milk in microwave in a 2 cup measure, add the eggs to the warm milk and stir briskly with fork. Add the egg/milk mixture to the hot pudding; mix in thoroughly and cook another 2-4 minutes, stirring constantly, or to desired thickness. Remove from heat, and stir in butter and vanilla. Serve warm with cinnamon and nutmeg, if desired.