Start by braising the duck legs in the stock. Salt the legs well and arrange in a Dutch oven or other heavy lidded pot. Pour in the stock and cook at 300°F until the meat falls off the bone, which could take anywhere from 90 minutes to 3 hours, depending on how old your birds are. When the meat is done, pull it off the bone and reserve. This step can be done up to several days in advance.
Make the crust. Mix the flour and salt in a large bowl and make a well in the center. Melt the lard and butter or duck fat in a small pot, then pour it into the well. Pour in the warm water and beaten egg and mix into a dough. Knead it by hand for 3 to 4 minutes, until it forms a cohesive ball. Chill in the fridge for at least 1 hour. This dough can be made a day in advance.
Make the pie. Cut off about 1/4 of the dough and set aside. Roll out the rest with a rolling pin until it is somewhere between 1/8 and 1/3 of an inch thick. It is supposed to be a thick crust you can manhandle, so don't get dainty. Make sure the dough can fit into your springform pan and up the sides. This usually determines how thick you roll the dough out. Press the dough into the springform pan so it reaches well up the sides, ideally to the top.
Preheat the oven to 350°F. Mix all the meats well with the herbs and spices. Fill the pie with the mixture.
Roll out the remaining dough to about the same thickness as the rest of the crust. Paint the top of the crust with the beaten egg and lay the rolled out dough on top. Seal by crimping the edges. Cut a hole in the center of the top about 1/2 inch in diameter.
Bake the pie at 350°F for 30 minutes. Lower the heat to 325°F, and bake for another 1 hour and 15 minutes.
Take the pie out of the oven and release the springform sides carefully. Paint the whole pie with the remaining beaten egg and bake for another 15 to 20 minutes, or until golden.
Remove the pie from the oven and let it rest. There is a chance that the meats may settle inside the pie, leaving a big under the top. This does not always happen. If it does, you can either leave it alone -- the pie will still taste fine -- or you can fill the gap with some warmed duck demi-glace (or any demi-glace). It must be gelled because the liquid will need to set up as a solid when the pie is chilled.
Leave the pie at room temperature for several hours, until cooled. Then store in the fridge. Serve chilled or at room temperature with homemade mustard and pickles.
Notes
Note that prep time does not include the time it takes to braise your meats beforehand.