Preheat the oven to 375º (180ºC.)
1. In a medium-sized saucepan, warm the butter, lime juice, sugar, zest, and salt.
2. In a separate bowl, whisk together the eggs and the yolks.
3. When the butter has melted and the mixture is warm, gradually pour some of the warm lime juice mixture into the eggs, whisking constantly. Scrape the warmed eggs back into the saucepan and cook the mixture over low heat.
4. Stir the mixture constantly over low heat, using the whisk, until the filling thickens and begins to resemble soft jelly. Do not let it boil.
(For the intrepid, you can do this step in a large bowl set over a saucepan of simmering water instead of over direct heat.)
5. Remove from heat and scrape the filling into the pre-baked tart shell.
6. Bake for 10 minutes.
7. To make the meringue (see Note for alternative method), whisk together the egg whites, sugar, and salt in the bowl of a stand mixer. Set the bowl over a pan of simmering water, and whisk it as it heats, checking it with an instant-read thermometer.
8. Once it reaches 140F (60C), transfer the bowl to the standing mixer and beat at high speed until cool, scraping down the sides once of the mixer bowl, midway during mixing, and add the vanilla. Whip until the meringue is light and fluffy.
9. Heat the broiler and move the oven rack to the top third of the oven.
10. Scrape the meringue into a pasty bag fitted with a star tip and pipe a ring around the perimeter of the tart. Or spread in a ring around the tart with a spatula.
11. Pop the tart under the broiler, watching carefully, as it will brown quickly. When the top begins to darken, remove the tart from the oven and cool completely before slicing.