Preheat the oven to 350 degrees F.
One at a time, hold the tortillas over the stovetop burner (heated to medium heat) to brown slightly, about 30 seconds per side. Set the warmed tortillas aside.
In a bowl, mix together the cumin, chili powder, Cajun spice, salt and pepper. Sprinkle both sides of the chicken breasts with the spice mix.
Heat the oil in a skillet over medium heat and cook the chicken on both sides until done in the middle and the juices run clear, 4 to 5 minutes per side. Set aside on a plate to cool, then shred finely with a fork.
Throw the onions into the same skillet, stir them around and cook until deep golden brown and caramelized, 4 to 5 minutes; set them aside on a plate. Pour the enchilada sauce into the skillet and reduce the heat to low, allowing it to warm through.
To assemble the enchiladas: Pour 2 cups of the sauce into a 9-by-13-inch casserole dish and leave the rest in the skillet. Dip a tortilla into the sauce in the skillet, then lay it on a baking sheet or plate. Sprinkle some cheese down the middle, followed by some chicken and finally, some of the caramelized onions. Roll it up tightly, then place it seam-side down in the casserole dish. Repeat with the rest of the tortillas. Pour the rest of the sauce over the enchiladas, then sprinkle on the rest of the cheese. Give it a final sprinkling of chili powder.
Bake until hot and bubbly, 30 minutes. Let it sit for 15 to 20 minutes before serving.
Serve the enchiladas topped with a dollop of sour cream, a sprinkle of diced tomatoes, a sprinkling of cilantro and a wedge of lime.