MAKE THE SAUCE: In a microwave-safe bowl, cover the chiles with the hot water. Microwave at high power for 2 minutes, until softened. Using a slotted spoon, transfer the chiles to a blender. Add 1 cup of the chile soaking liquid and the onion, garlic, cumin, coriander, and oregano and puree.

In a large saucepan, heat the olive oil over moderately low heat. Add the tomato sauce and the water. Strain the red chile sauce through a fine mesh sieve into the saucepan, pressing and scraping to remove the skins and any stray seeds. Cook, stirring occasionally, until the sauce is slightly reduced, about 15 minutes. Season with salt and pepper. Spoon about 3/4 cup of the sauce into the bottom of 2 shallow baking dishes.

MEANWHILE, MAKE THE ENCHILADAS: Preheat the oven to 350 degrees. In a large skillet, heat the olive oil. Add the onion and cook over moderate heat, stirring, until lightly browned, 10 minutes. Add the broth and cook, stirring, until the onions are very soft and the broth has evaporated, 10 minutes longer. Transfer to a bowl; let cool. Stir in the chicken, cumin, cilantro, and half of the shredded cheese. Season with salt and pepper.

Wipe out the skillet and heat 1/4 inch of vegetable oil in it. Add the tortillas 1 at a time and fry over low heat just until pliable, about 10 seconds each. Transfer to a paper towel-lined baking sheet and pat dry. Spoon 1/4 cup of the chicken filling onto each tortilla and roll up into a tight cylinder. Place the cylinders in the baking dishes, seam side down. Spoon the remaining chile sauce on top, spreading it to cover the enchiladas. Sprinkle with the remaining cheese.

Cover the enchiladas with foil and bake for 45 minutes, until heated through and bubbling; remove the foil halfway through cooking. Let the enchiladas cool for 10 minutes. Serve with chopped red onion, cilantro, hot sauce, and sour cream.

The dish can be prepared through Step 4 and refrigerated. Return to room temperature before baking.