1 small leek, thinly sliced, rinsed well, white and light green parts only (about 3/4 cups) 4 ounces
Make the dough:
Combine flour and salt in the bowl of a food processor. Pulse quickly to combine. Add the cubed butter and pulse until only pea-sized lumps of butter remain.
In a small separate bowl, whisk together the egg and apple cider vinegar. Add them slowly to the bowl of the food processor, pulsing to combine. The mixture will still be quite dry; add the ice water slowly as you continue pulsing until the mixture holds together when you hold a chunk and squeeze it between your fingers.
Divide the dough in half and flatten each half into a disk. Wrap in plastic wrap, and refrigerate for 1 hour or up to 2 days.
When ready to shape the tarts, remove the dough from the refrigerator and roll out each piece on a lightly floured work surface into a rectangle about 1/8-inch thick; trim the dough so it's an even 9x12 inches. Repeat with the second disk of dough.
Cut each piece of dough into thirds so you form nine 3x4-inch rectangles. Lay nine of the rectangles down on a greased or parchment-lined baking sheet. Refrigerate all pieces of dough until ready to assemble the tarts.
Make the filling and assemble the tarts: To make the filling, whisk together the Melville, egg, salt, chives, and parsley.
In a small saucepan over medium heat, warm the olive oil and add the sliced leek. Cook until soft and fragrant, 7 to 9 minutes. Set aside.
Mix the leeks into the herbed Melville mixture. Spread a heaping teaspoon of leek-Melville mixture on top of one of the dough rectangles and lay approximately ½ ounce of lobster mea on top.
Brush the edges of the tart with beaten egg, and place a second rectangle of dough on top of the first. Use your fingertips to press firmly around the edges, sealing the dough. Next, use the tines of a fork to press gently down around the edges, further sealing the pastry. Repeat with remaining tarts.
Brush the tops of each tart with beaten egg and prick a few times with a fork. Freeze for 30 minutes to 1 hour before baking. During this time, preheat the oven to 375°F.
Bake for 25 to 28 minutes, or until tarts are light golden-brown. Cool for at least 20 minutes before enjoying. Serve slightly warm or room temperature.