For the filling: Toast the chiles on both sides in a dry skillet over medium heat until pliable, about 3 minutes. Transfer to a large bowl and cover with boiling water; let stand until soft, about 30 minutes.
Reserve 1 cup of the soaking water and drain the rest. Puree the chiles, reserved soaking liquid, garlic, onion, oregano and cumin in a blender until smooth.
Heat the oil in a medium skillet over high heat until hot. Add the chile puree and fry, stirring (be careful, the mixture may splatter), until very thick and fragrant, about 5 minutes. Stir in the chicken broth and 1 teaspoon salt, then add the chicken. Bring to a boil, then reduce the heat to medium low and simmer, partially covered, turning the chicken occasionally, until cooked through, about 35 minutes.
Transfer the chicken to a cutting board. Increase the heat to medium high and simmer, stirring occasionally, until the sauce thickens, about 15 minutes. Shred the chicken into small pieces and then coarsely chop. Transfer to a bowl and add enough of the sauce to coat it well; season with salt. Reserve the remaining sauce to serve on the side. Refrigerate both until ready to use.
Soak the corn husks in a large bowl of hot water until pliable, about 1 hour. Fill a large pot with 1 inch of water and place a steamer insert inside.
For the masa: Mix the masa harina with 2 cups of the chicken broth, kneading by hand to incorporate. Cover and let stand for 20 minutes.
Beat the lard, baking powder and salt in a large bowl with an electric mixer on high until fluffy, about 2 minutes. Add the masa harina mixture in 2 batches and beat very well to combine. Add the remaining 3/4 cup chicken broth and beat until lightened and fluffy, about 3 minutes. Form a small ball of masa and see if it floats in a glass of water. If it does not, continue to beat the masa for a few minutes until the masa floats. Cover with plastic and let sit for 30 minutes.
Assemble the tamales: Drain the corn husks and wring them dry.
Lay a corn husk on a work surface, smoother side up, with the wide part of the husk closer to you. Spoon about 3 tablespoons of masa onto the bottom half of the husk and use an offset spatula to spread in a thin, even layer (leaving the top half of the husk bare). Spoon a heaping tablespoon of the chicken mixture onto the masa. Then roll the husk up to enclose the filling. Fold the narrow side of the tamale under and arrange the tamale, open-side up, in the steamer insert. Repeat with the remaining masa, husks and filling.
Bring the water in the pot to a boil over high heat, then reduce the heat to medium, cover the pot and steam until the masa is cooked through and separates easily from the husk, about 1 hour 20 minutes. Transfer to a platter and let stand for 10 minutes before serving.
Warm the reserved sauce and serve on the side.