Fill a large saucepan with lightly salted water and bring it to a boil. Add the lobster meat, reduce the heat to low, and cook 10 minutes until the meat is opaque. Drain and set aside to cool.
Place 4 tablespoons butter in a large saucepan and place it over medium heat. Allow the butter to melt and add the onion, carrots, and tarragon. Stir to coat, lower the heat, and let cook for 20 to 30 minutes, stirring occasionally. Add the peas in the last 5 minutes of cooking.
Meanwhile, place the diced potatoes in a small saucepan and cover with water. Add a little salt and place over high heat. Bring to a boil, reduce the heat and let simmer until the potatoes are fork tender (about 15 minutes). Drain the potatoes and add them to the vegetable mix. Chop the lobster meat and add to the veggie mix as well. Set aside.
For the bechamel:
Place the milk in a medium saucepan along with the bay leaves and white pepper and scald (which means heat it but don't let it boil!) Add the lobster base and whisk to incorporate. Melt the butter over medium heat in a medium saucepan. Whisk the flour in and cook for 1 minute. Add the wine and whisk until the mixture thickens. Slowly add the milk, whisking constantly, until the bechamel is thick and creamy. Remove bay leaves. Add the bechamel to the lobster mixture and stir well.
For the crust:
Place the flour in the bowl of a food processor along with the salt. Add the butter bits and pulse until the mixture resembles coarse crumbs. Pour into a bowl and add the ice water a little at a time, stirring well with a fork. Transfer the dough to a well floured surface and knead just until it all comes together (don't work the dough too much or it'll get tough). Divide the dough in half and flatten each ball into a disk. Wrap each disk in film and refrigerate for 1/2 hour. Remove from the refrigerator and roll each disk out on a lightly floured surface until it is about 12 inches in diameter. Line a 9-inch pie plate with the first disk, ladle the lobster mix into the crust and brush the edges of the dough with the egg wash. Top with the second disk. Trim the excess dough and crimp decoratively. Cut vent holes in the dough and brush the top with the egg wash. Bake for 30 to 40 minutes until the crust is a golden brown. Remove from heat and let cool for 20 minutes before cutting and serving.