Place currants in a medium stockpot over medium heat. Cook, crushing with a potato masher, until berries are completely soft, about 15 minutes.
Strain through a fine-mesh sieve. Do not force fruit through strainer; discard pulp. There should be about 1 cup juice. Measure and set aside 1/2 cup juice. Reserve remaining juice for another use.
Combine 1/2 cup currant juice, egg yolks, and sugar in a medium saucepan. Cook, stirring constantly, until the mixture thickens and coats the back of a wooden spoon, 10 to 12 minutes. Do not allow the mixture to boil.
Remove saucepan from heat. Add butter, one piece at a time, stirring with the wooden spoon until smooth.
Strain mixture through a fine-mesh sieve into a medium bowl. Lay a sheet of plastic wrap directly on the surface of the curd to avoid a skin from forming; wrap tightly. Let cool; refrigerate until firm and chilled, at least 1 hour.