Blend the flour and salt in a mixer with a paddle attachment or in a food processor, then add the butter just until the mixture resembles coarse crumbs. Add enough water for the dough to come together, then shape it into a disk and wrap with plastic. Chill the dough for 15 minutes.
Roll the dough into a 10-inch circle and place it on top of a 9-inch tart pan with removable rim. Shape the dough into the edges, the dough should stand about 1/4 inch above the rim and is about 1/4 inch thick. Prick the bottom with a fork 6 times, then freeze for 20 minutes. While the tart dough rests in the freezer, preheat the oven to 425 F. Save left over dough to patch the tart shell.
Place the frozen tart shell on a baking tray and bake until it's light in color, about 25 minutes. Check after 15 minutes and deflate any swollen pockets of dough with a prick from the tip of a knife. Remove the tart from the oven when ready, mend any holes by gently pressing in small pieces of left over dough. Cool on a wire rack and set aside.
Potato Leek, Goat Cheese, and Parsley Puree Filling
Preheat oven to 400 F. Keep the prebaked tart shell on its baking tray.
Slice the leeks and prebaked potatoes into thin rounds and place them in a bowl of cold water to remove dirt. Strain.
Heat the butter and olive oil over medium heat in a large skillet, add the leeks and potatoes and cook until tender, 10 to 15 minutes.
Prepare the parsley puree. Place all ingredients in a food processor or blender and process until smooth, stopping to scrape down the sides of the container if necessary. Taste and adjust the seasoning, if necessary. Set aside.
Toss the leek and potato mixture in a medium bowl with cup of parsley purée. Season with salt and pepper to taste.
Beat the goat cheese with the egg until smooth, then stir in the milk, crème fraîche, pinch of salt, and black pepper. Grate the Parmesan cheese onto the bottom of the tart shell. Pour the custard over the cheese layer.
Bake until the custard is golden and set in the middle, about 30 minutes. Sprinkle parsley leaves on top. Remove the tart from the pan and transfer to a plate. Serve warm.