Place the sugar in a medium bowl and zest the lime into it. Rub the zest and sugar together between the palms of your hands.
Combine the lime juice, eggs, egg yolk, butter, and the zested sugar in a medium nonreactive saucepan, and whisk to blend. Set the pan over medium heat and cook, whisking constantly for 3 to 5 minutes, until the mixture begins to thicken.