Preheat the oven to 400°F.
To make the pastry, in a food processor, combine the flour and granulated sugar and pulse a few times to mix. Add the butter and pulse until the mixture resembles fine crumbs. Add the egg yolk and pulse until blended. Add 3 tablespoons of the ice water and pulse just until the dough clings together, adding up to 1 tablespoon more water if needed.
Pinch off one-fourth of the dough and press it evenly onto the bottom and up the sides of a 4½-inch tartlet pan with a removable bottom. Repeat to line three more 4½-inch tartlet pans. Place the pans in the freezer for 10 minutes to firm up the pastry, then arrange them on a rimmed baking sheet. Line the shells with aluminum foil and use pie weights or dried beans to weight the shells. Bake the pastry shells until set but still pale, about 10 minutes. Remove from the oven. Reduce the oven temperature to 350°F.
In a large bowl, whisk together the eggs until light and fluffy. Add the brown sugar, corn syrup, melted butter, Cognac, salt, and vanilla and whisk until smooth. Stir in the nuts.
Divide the filling evenly among the pastry shells. Bake until the top is browned and the filling is set, 20–25 minutes. Let cool on a rack to room temperature, remove from the tartlet pans, and serve.