Peel, core, and dice the apples. Seed and chop the kumquats. Combine all the fruit with the brown sugar, half the granulated sugar, and the whiskey. Cover and refrigerate for at least 2 hours, preferably overnight. Wait until this step is done before preparing the crust.
Combine the flour, sugar, cardamom, and salt in a large mixing bowl. Using a pastry cutter or stand mixer (or two knives if you're a masochist), cut the butter into the dry ingredients. Continue until the pieces are no larger than peas.
In a small bowl, combine the egg yolk, vanilla, and cold water. Whisk together, then add to the crust mixture. Mix together until just combined. Form the dough into a ball and chill for 1 hour.
Remove the dough from the refrigerator and let it warm up for 5 minutes. Lay out a large sheet of wax paper or parchment paper and roll the dough into a ball on top of it. Roll the dough out into a circle large enough to fit in a 12 inch tart tin. The dough will be quite thin - about 3-4 mm (~1/8 inch).
Carefully lift the dough and transfer it to a 12 inch tart tin. Press the sides against the tin and trim any excess. The lip of the crust should come up only about 1 cm (~1/2 inch) or so.
Preheat the oven to 350 F. Prick the surface of the tart crust all over with a fork, then place in the oven to blind bake for 25 minutes, or until it is firm and golden. If air gets trapped beneath the tart, you can gently shake the tin to let it out. The final dough should be firm, but still a little bit pliant while warm. Remove the crust from the oven and set it aside to cool. Leave the oven on, unless you plan on waiting a while before making the filling.
Note that if you choose to use baking weights, you may want to extend the cooking time slightly, and be sure to remove them with a little time left to allow the crust to brown.
Melt the butter for the filling in a large pot over medium heat. Transfer the fruit/sugar/scotch mixture and the flour to the pot and stir. While the mixture is heating, whisk together the remaining sugar and the eggs. Once the apples are beginning to soften and the mixture is quite warm, slowly pour the egg and sugar mixture into the pot while stirring to combine. Once the mixture is evenly mixed, pour it into the prepared tart crust. Take care not to overfill the tart -- if you have extra filling, refrigerate it and use it for something else. Transfer the tart to the oven and bake for 20-25 minutes. The tart should be set, but will remain soft even after cooling.
Allow the cooked tart to cool at room temperature, then transfer to the refrigerator for at least 2 hours before adding caramel and serving.