Toast the dried chiles in a dry skillet over medium heat for 30 seconds per side until fragrant. Transfer to a bowl and cover with 2 cups of hot chicken broth; soak for 15 minutes until softened. In a blender, combine soaked chiles with the soaking liquid, garlic, onion, oregano, cumin, and salt. Blend until completely smooth. Heat oil in a saucepan over medium-high heat. Pour the sauce through a fine mesh strainer into the hot oil (it will sizzle). Simmer for 10 minutes, stirring occasionally. Add sugar if desired. Adjust consistency with remaining 1 cup broth as needed—sauce should coat the back of a spoon.
In a large bowl, combine shredded chicken, 1 cup Monterey Jack cheese, onion, cilantro, cumin, salt, and 1/2 cup of the prepared enchilada sauce. Mix until well combined. Taste and adjust seasoning.
Preheat the oven to 375°F (190°C). Spread 1/2 cup of enchilada sauce on the bottom of a 9x13-inch baking dish. Warm the corn tortillas briefly in a dry skillet or microwave (wrapped in damp paper towels) until pliable, about 20 seconds. This prevents cracking when rolling.
Place a heaping 2 tablespoons of filling down the center of each tortilla, roll tightly, and place seam-side down in the prepared baking dish. Repeat with all 12 tortillas, nestling them snugly. Pour the remaining sauce evenly over the top, spreading to cover all enchiladas. Sprinkle with the 2 cups of shredded cheese.
Bake uncovered for 20-25 minutes, until the cheese is melted, bubbly, and slightly golden. For a crispier cheese top, broil for the final 2 minutes, watching closely. Remove from oven and let rest for 5 minutes before serving.
Drizzle with Mexican crema, sprinkle crumbled queso fresco and fresh cilantro over the top. Serve immediately with lime wedges on the side.