Place chicken and quartered onion in a pot (that has a lid) and add enough water to cover by a couple of inches. Bring to a boil, then lower heat to LOW, cover, and gently simmer for 20 minutes. Use a spatula to carefully remove the chicken from the broth and place on a rack to cool.
Save about 1 cup of the broth. Remove the onions and save the remaining broth for another use.
When cool enough to handle, remove the skin from the chicken and then pull the meat from the bones. Use your fingers, or a couple of forks, to shred the meat. Set aside.
In a large skillet, heat the oil over medium heat until shimmering. Add the chopped onion and cook until soft and translucent, about 4 to 6 minutes.
Add the garlic and cook for another 30 seconds. Add in the chilis and cook for another minute, stirring frequently.
Add the chicken, taco seasoning, and flour, stir until the chicken is completely coated. Add about ¾ cup of the reserved broth. Stir to incorporate. Continue cooking until the chicken is warm and a slight sauce has formed (there won't be a lot of sauce). Set aside.
In a cast-iron skillet, add a few tablespoons of vegetable oil and heat over medium heat until shimmering. One at a time, carefully add a corn tortilla to the hot oil, after about 1 to 2 seconds, use tongs or a fork to flip the tortilla, and cook for another 1 to 2 seconds (no more than 4 to 5 seconds total). Carefully transfer the tortilla to a cutting board lined with paper towels. Continue this process until all tortillas are softened.
If using a deep fryer, heat it up to 350°F. If using your cast-iron skillet, add more oil to it until it's about 1 to 2-inches deep. Heat to 350°F, using a candy thermometer to reach the correct temperature, if possible.
Meanwhile, place about 2 tablespoons of the chicken filling down the middle of each softened tortilla. Top with another couple of tablespoons of the shredded cheese. Roll the tortilla tightly and use a toothpick (1 or 2) to affix the edge of the tortilla to the body. (See Photo and/or video for reference).
Working in batches, place the rolled tortillas in the deep fryer (preferably with a basket) of cast-iron skillet with hot oil, and cook until crispy and lightly browned, about 2 to 3 minutes. Remove, keep warm in a low-temp oven (200°F) and repeat with remaining tortillas.
If desired, serve the flautas warm over a bed of shredded lettuce, garnished with Mexican crema, pico de gallo, and salsa for dipping.