Toast the guajillo and ancho chiles in a dry skillet over medium heat for about 2 minutes until fragrant, then place them in a bowl and cover with 2 cups of hot chicken broth. Let soak for 15 minutes until softened.
Transfer the softened chiles and soaking liquid to a blender, add the garlic, cumin, oregano, and remaining 1 cup of chicken broth, then blend until completely smooth. Strain the sauce through a fine-mesh sieve and season with salt to taste.
Pour the chile sauce into a wide, shallow bowl and heat the vegetable oil in a large skillet over medium-high heat.
Quickly fry each tortilla in the hot oil for about 10 seconds per side just to soften, then immediately dip it into the warm chile sauce to coat both sides.
Fold each sauced tortilla in half like a taco, fill with shredded chicken, and place on serving plates, overlapping slightly.
Spoon additional chile sauce over the enchiladas, then top with crumbled queso fresco, diced white onion, a drizzle of Mexican crema, and fresh cilantro before serving immediately.