Heat the pineapple juice on medium until it simmers. Simmer, stirring occasionally, until it is reduced in volume by about half.
Meanwhile, cut the butter into tablespoon-size pieces and separate your eggs if you haven't already.
Reduce the heat to medium low and add in the lime juice, sugar, eggs and egg whites. Stir or whisk the mixture constantly.
When the mixture starts to thicken, add the butter one piece at a time and stir until it disappears before adding the next piece.
After the butter is incorporated, you should be able to leave a trail down a coated spatula or spoon with your finger. If not, continue cooking on low until it reaches this consistency.
Set a strainer on top of a mixing bowl. Take the mixture off of the heat and pour it through the strainer. You can tap the strainer and scrape the underside of it with a spatula. Try not to push down on any egg lumps that remain in the strainer itself.
Chill for at least 4 hours or overnight until set. Pour into containers and seal tightly.
It will keep for about a month in the refrigerator, or it also freezes well in a freezer-safe container.