In the top of a double boiler, combine the lemon zest, eggs, sugar, and lemon juice, and whisk until blended. Over very low heat, simmer, stirring occasionally, until the mixture is thick enough to coat the back of a wooden spoon, 6 to 8 minutes.
Stir in the butter until melted and smooth. Immediately strain through a fine-mesh sieve. Chill the lemon butter, uncovered, in the refrigerator until cool, 30 to 40 minutes.
Store the lemon butter airtight in an impeccably clean container in the refrigerator. It will keep for up to 1 week.