In a small saucepan, bring 1 cup of the water to a boil over high heat. Stir in bulgur and remove pan from heat. Let sit 1 hour, then drain in a fine sieve; you should have about 1 cup.
Meanwhile, preheat oven to 350. Spread walnuts on a baking sheet or in a shallow pan. Bake, stirring once or twice, until lightly browned and fragrant, 8 to 10 minutes. Let cool.
To bone the game hens: Start from the neck opening. Without tearing the skin, cut around the wishbone and remove it. Cut at the shoulder joints and push with your fingers to separate the breast meat from the bone. Remove the rib cage and breastbone but leave the legs and wings intact. (Reserve the bones to make stock or soup.)
In a large frying pan, heat 1 tablespoon of the oil over medium heat. Add leek and onion; cook, stirring occasionally, until softened but not browned, about 3 minutes. Add garlic, jalapeño to taste, apple, thyme, ½ teaspoon of the salt, the pepper, and bulgur. Cook, stirring, until any additional moisture from the bulgur is absorbed. Let cool completely. Stir in walnuts.
Preheat oven to 425. Stuff game hens with the bulgur-walnut mixture. Use a skewer or string to enclose the stuffing.
Spread the remaining 1 teaspoon oil in the bottom of a small roasting pan. Place hens in pan and sprinkle with the remaining ¼ teaspoon salt. Roast until golden brown on the outside and cooked throughout, about 40 minutes, basting every 15 minutes with the juices and fat in the bottom of the pan.
Remove hens to a plate. Pour out and discard any fat that has accumulated in the roasting pan. Add the remaining ½ cup water to the pan and bring to a boil, stirring to release any browned bits in the pan.
Remove skewers and cut hens in half. Arrange any extra stuffing on 4 warmed plates. Set hens on top and spoon over cooking juices. Garnish with sautéed apple slices.