Heat oil in a large skillet to 360°F, enough to cover the flautas about halfway.
Combine the shredded chicken, green chilis, taco seasoning, salsa, and cheddar cheese in a bowl.
Warm the tortillas in the microwave, wrapped in a towel, for about 45 seconds, until pliable.
Distribute approximately 1/4 cup of the chicken mixture onto each tortilla, roll tightly, and secure with a toothpick if necessary.
Fry the flautas in batches, turning occasionally, until evenly golden brown, about 4-5 minutes.
Drain on a paper towel-lined plate and let cool for 5 minutes before serving.