Preheat an oven to 350°F.
To prepare the tart dough: Place the flour, sugar, salt and butter in the bowl of the food processor and process until the mixture resembles fine breadcrumbs. Sprinkle in the water and mix until the mixture forms a ball round the blade. Turn out on to work surface and shape into a rectangle. Wrap in plastic wrap and chill for 30 minutes.
To prepare the tartlets: Lightly flour a work surface and roll pastry out to 1/8-inch thick. Using a 2 1/2-inch round cutter, cut out rounds. Gather trimmings on top of each other (do not squash together) and roll out remainder to 1/8-inch thick, cut out pastry rounds. Use cut pastry to line the cavities of a 24-cup mini muffin pan. Press pastry gently into the base. Repeat to fill second pan, you will not line all of the cavities. Put the muffin pans in the refrigerator to chill the pastry while you prepare the filling.
To prepare the filling: Place the butter in a medium saucepan and melt on the stove over medium heat. When melted, add the sugar and stir until smooth, about 2 minutes. Remove the saucepan from the heat and whisk in the corn syrup, vanilla bean paste, rum and salt. In a large measuring cup with a spout, whisk the eggs until frothy and then whisk in the corn syrup mixture.
Evenly divide the pecans and chocolate between the muffin cavities and pour the corn syrup mixture over the pecans; transfer to the oven and bake until the filling is set, 20 to 25 minutes. Allow to cool completely before serving.