Heat 1 teaspoon vegetable oil in a skillet over medium heat. Stir in the bell pepper and onion; cook and stir until the onion has softened and turned translucent, about 5 minutes. Increase heat to medium-high and stir in the chicken breast. Cook and stir until the chicken breast is no longer pink in the center, about 10 minutes. Shred the chicken using two forks. Stir in the taco seasoning and water. Simmer until the liquid has evaporated, stirring occasionally, about 10 minutes. Stir in the Cheddar cheese, then remove from heat, and set aside.
Lightly brush each corn tortilla will a layer of salsa. Spread about 2 tablespoons of the chicken mixture in a line along the bottom edge of the tortilla. Tightly roll the tortilla into a cylinder, and secure the ends with one or two toothpicks. Repeat with remaining tortillas.
Heat 2 cups vegetable oil in a large skillet to 375 degrees F (190 degrees C).
Fry the flautas in the preheated oil, no more than 4 at a time, until golden and crisp, about 4 minutes. Drain the flautas on a paper towel-lined plate. Discard toothpicks and top finished flautas with remaining salsa.