Preheat the oven to 350 degrees F. Make the crust: Lightly coat a 9-inch fluted tart pan with a removable bottom with cooking spray. Spread the pecans on a baking sheet and roast until golden brown, 8 to 10 minutes; let cool. Transfer 1/2 cup of the pecans to a food processor; add the flour, sugar and kosher salt and pulse until the nuts are finely ground. Add the butter and pulse until it is in pea-size bits. Pulse in the egg yolk and vanilla until large clumps form.
Transfer the dough to the prepared pan and press it into the bottom and up the sides. Prick the bottom all over with a fork, then freeze until firm, about 20 minutes.
Line the crust with foil and fill with pie weights or dried beans. Bake until set, about 25 minutes, then remove the foil and weights and continue baking until golden, 10 to 15 more minutes. Transfer to a rack and let cool completely.
Make the caramel for the filling: Combine the sugar, 1/4 cup water and the kosher salt in a medium saucepan over medium heat, stirring just until the sugar dissolves. Cook, swirling the pan but not stirring, until amber, 8 to 10 minutes. Brush any sugar crystals off the side of the pan with a wet pastry brush. Remove from the heat and carefully whisk in the heavy cream, butter and vanilla (the mixture will bubble). Return to medium heat and simmer, whisking constantly, until thick enough to coat a spoon, 3 to 4 minutes. Let cool 5 minutes.
Gently whisk the eggs and flour in a large bowl, then whisk in the caramel. Place the crust on a baking sheet, arrange the remaining 1 1/4 cups pecans in the bottom and add the filling. Bake until set around the edge, 30 to 35 minutes. Transfer to a rack to cool.
Put the chocolate in a microwave-safe bowl and microwave in 30-second intervals, stirring, until smooth. Drizzle over the tart and sprinkle with sea salt. Let the chocolate set before slicing.
Photograph by Kana Okada
Recipe courtesy Food Network Magazine