For the hen: Put the hen in a 1 gallon resealable plastic bag with the water, honey, and salt. Seal the bag and move around vigorously for 1 to 2 minutes to distribute the honey and dissolve the salt. Set the bag in a leak-proof container and refrigerate for 1 hour.
For the pastry: Heat the oven to 350°F.
Whisk together the flour and salt in a small mixing bowl and set aside. Combine the milk, water, and lard in a 3-quart saucepan over medium-high heat and bring to a boil, approximately 3 minutes. Remove the pan from the heat, add the flour all at once, and stir with a wooden spoon until combined and the mixture begins to form into a ball of dough.
Place the dough onto a floured work surface and knead until cooled, approximately 5 minutes. Divide the dough in half, and divide one of the pieces in half again. Set aside.
For the egg wash: Combine the egg and water in a small bowl. Set aside.
To build the pot pie: Remove the hen from the brine and pat dry. Use kitchen shears to cut away the skin from the breast of the hen. Place the dried fruit into the cavity of the hen and tie the legs with butcher's twine. Place the hen in the center of a piece of parchment and trace an oval approximately 1/2 inch larger than the hen, approximately 8 1/2 inches long by 6 1/2 inches wide. Remove the hen from the parchment and turn the paper over to avoid getting pencil on the dough.
Roll the two quarters of dough into ovals according to the template; this is the base and the top of the pot pie. Place one of the ovals in the center of the parchment. Set the other oval aside.
Roll the remaining half of dough into a 1/2-inch thick and 4-inch wide strip of dough that will go around the oval piece and surround the hen. Brush around the edge of the base pastry with egg wash. Place the thyme sprigs on the base, then top with the hen. Place the 4-inch wide strip of dough around the hen, forming a "wall" around the hen. Press the ends together. Press down to seal this to the base pastry. Sprinkle the hen with the allspice, black pepper, cumin, and paprika. Brush the top of the "wall" with egg wash. Place the remaining oval of pastry on top of the hen and crimp, attaching the top to the wall. Patch any holes with excess dough. Brush the entire pastry with egg wash.
Bake until the hen reaches an internal temperature of 150°F, about 1 hour and 15 minutes. Remove from the oven and rest 20 minutes before serving.