Make the chile sauce
Break open the dried chiles, discard the stems, and shake out the seeds. Depending on how brittle the chiles are, you can use your hands or use kitchen scissors. The chiles contain a natural chemical that can irritate your skin or eyes, so be careful. If your skin is particularly sensitive, wear rubber gloves. Be sure to wash your hands thoroughly with soap and water when you have finished.
Place a large frying pan over medium-high heat, add the chiles, and toast them by pressing them flat onto the hot surface with a metal spatula. Once they change color and you can start to smell them, they are ready. Don’t toast them too long or they’ll scorch. Transfer the chiles to a medium bowl, add hot water to cover, and let soak until quite soft, at least 30 minutes.
While the chiles are soaking, in a medium frying pan, heat the oil over medium high heat. Add the onion, season with 1/4 tsp. salt, and cook, stirring frequently, until the onion starts to sizzle. Then reduce the heat to low and continue to cook until the onion is very soft and sweet and is starting to caramelize, about 15 minutes. Add the garlic, cumin, and sugar and cook for another 30 seconds or so. Remove from the heat and set aside until the chiles are ready.
Drain the chiles, reserving the soaking water. In a blender, combine the chile and the onion mixture and process until pureed, adding enough of the broth to create a smooth puree and stopping to scrape down the sides of the blender. Now add the remaining broth and process, adding a little of the soaking water if needed to get a nice pouring consistency reminiscent of very thick canned tomato juice. Taste and adjust the seasoning with more salt or sugar if needed.
Make the enchiladas
Heat the oven to 250°F to keep the enchiladas warm.
To keep things from getting messy, set up your workstation like this: Pour the chile sauce into a shallow dish, such as a pie plate. If you’re right-handed, set this to the left of the stove and put the stack of tortillas within easy reach. Put a large baking dish—a 9-by-13-inch dish is a good size—to the right of the stove.
In a frying pan, heat 1 Tbs. of the oil over medium-high heat. Dip a tortilla into the chile sauce, coating on both sides and letting a little bit drip off, and then lay the tortilla in the hot oil. Watch out, as it will spit and sizzle. If there is room in the pan, coat a second tortilla and add it. Cook for 30 seconds, turn, and cook on the second side for a few more seconds until the tortilla is floppy and hot. Slip tongs or a long, thin metal spatula carefully under the middle of the tortilla, lift it carefully, and let any oil drip back into the pan. Transfer the tortilla to the baking dish, fold it in half, and then in half again to make a loose triangle.
Repeat with the rest of the tortillas, adding more oil to the pan as needed, and snugging the tortillas together in a neat single layer in the baking dish. Crumble the cheese evenly over the folded tortillas and pop them into the oven to warm back up, 5 to 7 minutes. In a small bowl, stir together the sour cream and lime juice and drizzle over the surface of the enchiladas. Serve right away.
The sauce can be refrigerated in an airtight container for up to 1 week.