
Prepare the frying oil. Fill a large sauté pan or deep skillet with 2 inches of oil and heat over medium-high heat until the temperature reaches 350°F. (To test, drop a small piece of tortilla in the oil. If it sizzles, it's ready.)
Make the chicken filling. In a large bowl, mix together the shredded chicken, cheese, onion, salt, and oregano until well combined.
Prep the tortillas. Wrap the corn tortillas in a damp paper towel and heat them in the microwave for 1-2 minutes, flipping the tortillas halfway through until all of them are warm and pliable. Place them in a tortilla warmer or a large resealable plastic bag to keep them warm during assembly.
Assemble and fry. Place a heaping spoonful of the shredded chicken onto each tortilla. Tightly roll up each tortilla and secure it closed by threading a toothpick along the seam.
Cooking in batches of 3 or 4, carefully add the flautas to the hot oil and fry for 1 ½ to 2 minutes per side until golden brown. Transfer them to a large plate or baking sheet lined with paper towels to soak up any excess oil, then remove and discard the toothpicks.
Make the salsa roja. Heat the oil in a large skillet over medium heat. Add the tomatoes, tomatillos, onion, and garlic, and cook until the vegetables start to soften and brown, about 10 minutes. Transfer everything to a large blender.
Add árbol chiles to the same skillet and sauté for 30-60 seconds until fragrant. Transfer it to the same blender along with the water, salt, and oregano. Blend until completely smooth and set aside.
Assemble the flautas ahogadas by arranging 4 flautas on a plate or low-sided bowl. Spoon ½ cup of the salsa roja on top, then garnish with a drizzle of Mexican crema, crumbled cotija or queso fresco, shredded lettuce, and chopped cilantro.