Adjust oven rack to the middle position and heat the oven to 400 °F (200 °C).
Combine the onion, oil, and 1/2 teaspoon salt in a large saucepan. Cover and cook over medium-low heat, stirring often, until the onions have softened, 8 to 10 minutes.
Stir in the garlic, chili powder, cumin, and sugar, and cook until fragrant, about 30 seconds. Stir in the tomato sauce and water, bring to a simmer, and cook until slightly thickened, about 5 minutes.
Nestle the chicken into the sauce. Reduce the heat to low, cover, and cook until the chicken is no longer pink in the center, and the thickest part registers 160 °F (70 °C) on an instant-read thermometer, 10 to 12 minutes. Transfer the chicken to a plate; set aside to cool. Strain the sauce through a medium-mesh strainer into a medium bowl, pressing on the onions to extract as much liquid as possible. Season the sauce with salt and pepper to taste.
Once the chicken is cool enough to handle, shred into bite-sized pieces. Toss together the shredded chicken, 1/2 cup of the enchilada sauce, 1 cup of the cheddar, the jalapenos, and cilantro, and season with salt and pepper to taste.
Stack the tortillas on a microwave-safe plate, cover with plastic wrap, and microwave on high until warm and pliable, 40 to 60 seconds. Spread the warm tortillas out over a clean work surface. Place 1/3 cup of the chicken mixture evenly down the center of each tortilla. Tightly roll each tortilla around the filling and lay them seam-side down in a 13 x 9" baking dish.
Lightly spray the tops of the enchiladas with vegetable oil spray. Pour 1 cup of the remaining sauce over the enchiladas to coat them thoroughly. Sprinkle the remaining cup cheddar down the center of the enchiladas. Cover the baking dish with foil and bake until the enchiladas are heated through, 20 to 25 minutes.
Remove the foil and continue to bake until the cheese browns, about 5 minutes longer. Serve, passing the remaining 1 cup sauce and the lime wedges separately.
There are 413 calories in 1 serving of Spicy Cheesy Chicken Enchiladas.