Follow the steps in order for the best result.
In a medium-sized saucepan or pot, combine the uncooked short-grain rice, fresh water, a pinch of salt, and the unsalted butter.
Bring the mixture to a gentle boil, then reduce the heat to low, cover the pan, and allow it to simmer for approximately 15 minutes.
Simultaneously, in a separate saucepan, gently heat the whole milk over medium-low heat until it reaches a scalding point (just before boiling, with small bubbles forming around the edges). Be careful not to let it boil over.
Carefully pour the hot, scalded milk into the rice mixture. If you're including raisins, now is the time to stir them in. Cover the pot once more and continue to simmer for an additional 15 minutes, or until the rice is tender and has absorbed much of the liquid.
While the rice is simmering, whisk together the large eggs and granulated sugar in a separate bowl until well combined and slightly frothy.
To temper the eggs, gradually add a small amount of the hot rice and milk mixture to the egg and sugar blend, stirring continuously and vigorously. This prevents the eggs from scrambling.
Pour the tempered egg mixture back into the pot with the remaining rice pudding. Continue to cook over low heat for about 5 minutes, stirring constantly with a wooden spoon or spatula, ensuring the bottom doesn't stick.
The pudding will thicken noticeably and should begin to gently bubble, indicating it's nearly ready.
Remove the pot from the heat and carefully transfer the creamy rice pudding into a serving bowl or individual ramekins.
Stir in the pure vanilla extract, ensuring it's evenly distributed throughout the warm pudding.
Allow the rice pudding to cool completely before serving. It can be enjoyed warm or chilled, garnished with a sprinkle of cinnamon or fresh berries if desired.