Place your cut butter into a heat-safe bowl and put a fine mesh sieve over top, set aside

In a medium pot, whisk together your lemon zest and juice, sugar, and eggs and place on the stove over medium-low heat. Whisk constantly until the mixture thickens, about 6-10 minutes, without letting it boil

Pour your mixture through the strainer into the bowl with the butter. Throw away any of the bits left in the strainer

Use a spatula to stir your mixture into your butter until smooth

Pour into a container and set aside to cool to room temperature before sealing and placing in the fridge to cool completely