In a medium bowl, toss the shredded chicken breast with the chipotle peppers and adobo sauce until well combined. Add more or less peppers and the adobo sauce to your liking.
To assemble, lay the tortillas on a work surface and fill the bottom third of each tortilla with 1/3 cup of the shredded chicken mixture and sprinkle with 1/4 cup of the shredded cheese. Roll the tortillas up tightly and secure the end with a toothpick inserted into the seam.
To fry the taquitos:
Pour enough oil in a heavy bottom fry pan or pot to cover 1/2 inch, or so enough oil reaches halfway up the sides of the tortillas when frying. Heat the oil to 350°F — use a candy thermometer to measure the oil without touching the bottom of the pan. If you don't have a thermometer, test the oil temperature by dropping a piece of the tortilla into the oil. If it sizzles it should be hot enough.
Add the filled tortillas to the hot oil and cook in batches so not to crowd the pan. Fry each side until golden brown, about 2 minutes each side. Transfer to a plate or cutting board lined with paper towels.
To bake the taquitos:
Preheat the oven to 425°F. Place the tortillas on a baking sheet lined with parchment paper or foil, and lightly brush the filled tortillas with canola oil.
Bake for 12-15 minutes or until golden brown and crisp then transfer to a plate.
To serve, remove the toothpicks from the taquitos and serve on their own or serve with avocado salsa verde, pico de gallo, sour cream or Mexican crema, shredded lettuce, and grated or crumbled cotija cheese.