Prepare the crust. Put the flour, sugar, salt, and butter in the bowl of a food processor and pulse until the mixture resembles coarse sand flecked with pea-sized pieces of butter. Add the eggs and process until the dough just comes together. Form the dough into a disk, wrap it in plastic wrap, and chill until firm, at least 1 hour.
Place the dough between two sheets of parchment paper and roll it out to form a 13” circle. Remove the parchment and transfer dough to a tart pan, pressing it into the corners so that the top of the crust is flush with the top of the pan. Cover the crust with plastic wrap and transfer to refrigerator to chill.
Heat the oven to 350°.
Prepare the filling. Combine the brown sugar and corn syrup in a small saucepan over medium heat, stirring to dissolve. Let the syrup to come to a boil, then turn off the heat and add the butter, stirring until well incorporated. Set aside briefly to cool. Whisk the eggs in a large bowl until smooth. Slowly pour 1/3 of the syrup mixture into beaten eggs, stirring continuously. Then, stir the tempered egg mixture into the remaining syrup and whisk to combine. Add the vanilla and salt.
Pour the filling into the chilled tart crust. Bake until the filling begins to thicken and set, about 30 minutes. Remove the tart from the oven and sprinkle with an even layer of pistachios, pressing the nuts gently into the filling with a rubber spatula. Return to the oven and bake until the nuts are lightly toasted and the crust is golden, about 5-10 minutes more. Let tart cool completely before cutting.