In a small pot, combine 2 cups water with honey and salt. Heat just until honey and salt are dissolved. Add 1/2 lb ice and stir. When completely cool, add game hen and soak 1 hour.
Preheat oven to 350°F.
Whisk together flour and salt. Set aside.
In a large pot, combine milk, water and lard. Bring to a boil. Turn off the heat and dump in the dry ingredients. Stir until it forms a loose dough.
Turn dough out onto a board, and as soon as it is cool enough, begin to knead. Knead until it forms a smooth dough, then set aside to cool for at least 5 minutes.
Using scissors or shears, cut through the skin along either side of the breast and remove the skin from over the breast. Stuff the cavity with the dried fruit, then tie the legs together with butcher's twine. Place bird on a sheet of parchment and trace around the bird, about 1/2" larger than the bird's shape. Set bird and parchment aside.
Cut dough in half, then cut one half in half again. Turn the parchment over so the pencil is on the underside. Using the heel of your hand, spread one of the smaller lumps of dough out to fill the shape you traced. Set aside, then repeat with the other small piece of dough; leave it on the parchment. Take the larger piece of dough, using bench flour as needed to prevent sticking, and roll out into a piece about 4" wide by 12" long by 1/2" thick.
Beat egg yolk and water together to form egg wash. Brush the base piece with egg wash. Scatter thyme sprigs over base, then top with the game hen. Place the long "wall" piece of dough around the game hen, crimping it to the bottom, then overlapping and pinching the two ends together. Sprinkle the breast of the game hen with the allspice, black pepper, cumin and paprika. Brush the top edge of the wall with egg wash, top with the remaining piece of dough and crimp to seal. Brush all exposed dough with egg wash.
Transfer parchment to a baking sheet, and place in preheated oven. Bake 1 hour 15 minutes.
Remove from oven and allow to rest 15 minutes before carving. Remove the top of the crust with a serrated or electric knife before extracting and carving the hen for service.