Whisk the sugar, 1 tablespoon lime zest, the lime juice and salt in a medium saucepan. Whisk the whole eggs and yolks in a small bowl, then whisk into the lime mixture.
Cook the mixture over medium heat, whisking constantly, until it¿s thick like pudding, 6 to 8 minutes. Remove from the heat and whisk in the butter a few pieces at a time until incorporated.
Strain the mixture through a fine-mesh sieve into a small bowl, pushing it through with a rubber spatula. Press plastic wrap directly onto the surface to prevent a skin from forming; let cool to room temperature. Stir in the remaining 1 teaspoon lime zest. Cover and refrigerate until completely set, at least 4 hours and up to 5 days.