In a bowl, combine the chicken and oregano. Season with salt (usually about ½ teaspoon, depending on how the chicken was cooked).
In a large (10-inch) skillet, heat the oil over medium until hot enough to make the edge of a tortilla sizzle sharply. With tongs, pick up 3 tortillas at a time, slide them into the hot oil, wait a few seconds for them to sizzle sharply, then quickly remove and drain on paper towels.
When the tortillas are just cool enough to handle, roll about 2 tablespoons of chicken tightly into each one, laying them seam side down once they are rolled. On about 3-inch wooden picks, skewer 3 taquitos together, making sure that the flaps are held in place on the inside.
In a medium bowl, mash the avocados with the salsa and cilantro. Taste and season with salt if necessary.
Return the oil to the heat. When hot enough to make the edge of a taquito sizzle sharply, fry the skewered taquitos in two batches, flipping after about 3 minutes, when the undersides are crispy. Drain on paper towels.
Strew the romaine over the bottom of a large rectangular platter. Line up the taquitos over the lettuce, spoon on the crema, then the guacamole, radishes and queso fresco.