Preheat oven to 450°F with a rack in the upper third. Finely chop the onion and 1 tablespoon of garlic. Coarsely grate all of the mozzarella on the large holes of a box grater. Cut chicken into strips, then into ½-inch pieces.

Transfer ½ cup of the chopped onions to a medium bowl, then stir in 2 tablespoons vinegar, 2 teaspoons sugar, and ½ teaspoon salt; set aside to marinate.

In a separate medium bowl, whisk taco seasoning and 1 tablespoon flour; slowly whisk in all of the tomato sauce, 1¼ cups water, 2 tablespoons of oil, and 2 teaspoons of vinegar; season to taste with salt and pepper.

Stack the tortillas, wrap in foil, and place on the upper oven rack until warmed through, about 5 minutes.

Meanwhile, heat 2 tablespoons of oil in a large skillet over medium-high heat. Add chicken, remaining chopped onions, and a pinch of salt. Cook until the onions are softened and chicken is cooked through, about 3 minutes.

Add the chopped garlic; cook until fragrant, about 30 seconds. Remove skillet from the heat, then stir in 1 cup of the sauce and half of the cheese; season to taste with salt and pepper.

Spread a ½ cup of the sauce into the bottom of a large baking dish. Spread the tortillas onto a work surface. Divide chicken filling among tortillas, then roll up each and place in the prepared baking dish, seam side down.

Pour the remaining sauce over the enchiladas, then sprinkle with the remaining cheese.

Bake on the upper oven rack until cheese is melted and sauce is bubbling, 10–15 minutes. Let sit for 5 minutes.

Serve chicken enchiladas topped with the pickled onions.