In a large bowl, vigorously whisk the eggs, egg yolks, sugar, and salt until smooth and pale-yellow.
In a separate bowl, whisk the flour, fresh-squeezed lemon juice, and vanilla extract until combined and the flour is dissolved.
Melt the unsalted butter in a medium pot. Add the lemon juice/flour mixture and cook, whisking constantly, until starting to bubble and thicken (about 2 minutes). Remove from heat and slowly pour the hot liquid into the bowl with the eggs. Whisk the eggs vigorously as you do so they don't curdle.
Once all of the hot liquid has been added to the eggs, whisk until the mixture is completely smooth, then pour everything back into the pot. Bring the curd to a boil, whisking constantly, until very thick (about 7-10 minutes). Note: Once the curd starts to boil it will thicken rapidly, if it's thickening too fast for you to whisk it and keep it smooth, lower the heat or take the pan off the heat for a minute and whisk until it smooths out.
Once it's thickened to the desired consistency (think custard or pudding consistency), immediately strain the lemon curd through a sieve and into a bowl. Use a whisk to help get it through the sieve, then use a rubber spatula or the back of a spoon to press the rest of it through. Use a different spatula or spoon to scrape the strained curd from the other side and into the bowl.
Pour the warm lemon curd into the pre-baked crust and use an offset or rubber spatula to spread it out smoothly and evenly. Bake the lemon tart for 15 minutes.
Take the tart out of the oven and allow it to cool completely at room temperature, then in the fridge for at least 2 hours before slicing and serving.