Fill bottom of double boiler with water, leaving enough room for water to simmer without touching the top pot. Bring to a gentle simmer over a medium-low heat.
Pour milk and rice in top of double boiler. Cover and nest into bottom of double boiler.
Cook for 45 minutes or until rice is done. Not all of the milk will be absorbed. Remove from stove top.
In a bowl, beat eggs until well-combined.
Temper the eggs by stirring a spoonful or two of the hot rice into beaten eggs. Mix quickly until completely blended. Once the eggs are slightly warmed, continue to add hot rice until about a third of the pot is now in the bowl with the eggs. This tempering will give you the rich flavor of the eggs without a scrambled egg consistency.
Pour egg and rice mixture back into the pot. Mix well.