Combine the flour and salt in the bowl of a food processor, pulse to combine. Add the cold butter and continue to pulse until the butter is incorporated and the texture of the dough looks like coarse breadcrumbs. Add the ice water and vinegar and process until the dough just comes together. Dump it out onto the counter and knead it a bit to bring it together. Divide the dough into two pieces, one somewhat larger than the other. The larger piece will be for the bottom crust, so approximately 2/3 to 1/3. Shape each piece into a disk and then wrap them separately in plastic wrap. Refrigerate for at least two hours or longer.

Combine the carrots, celery, onions and garlic in a food processor and pulse until coarsely and somewhat uniformly chopped.

In a large mixing bowl combine the chopped vegetables with the pork.

Place a dutch oven or deep saute pan on the stove over medium-high heat. Once the pot is hot add about 2 tablespoons of olive oil. Add the pork and vegetable mixture to the hot pan and sauté for 10 to 15 minutes until the pork is browned and the vegetables have released most of their moisture.

Add the tomato paste, canned tomatoes, thyme and wine. Cook for 20 minutes.

Using a ladle dip out 2 cups of the liquid from the pot and set aside for the sauce. Continue to cook the ragu until the remaining liquid has almost entirely cooked off, about another 20 minutes. Season well with salt and pepper and set aside to cool.

Preheat the oven to 375 degrees. Roll the larger disk of dough ints round large enough to fit a 9 inch pie pan. Fit the dough into the pan and fill with the pork ragu. Roll out the second piece of dough and place over the filling. Pinch the edges together all the way around and pole a hole in the top (decoratively or not, as you like) to allow steam to escape.

Use a pastry brush to brush the entire crust with the beaten egg, Bake the pie until golden brown, about 40-50 minutes. Remove from the oven and let cool at least 15 minutes before serving.

While the pie bakes, melt 3 tablespoons butter over medium high heat. Whisk in the flour and cook over medium low heat, stirring almost constantly, for about 5 minutes. Whisk in the reserved cooking liquid and simmer for about 20 minutes, then ad the remaining tablespoon of butter, salt and pepper. Whisk to incorporate. Keep warm.

Serve the pork pie warm with warm sauce.