In a medium pot, combine the chicken breasts, quartered onion, and garlic cloves. Add enough water to cover everything by an inch. Bring to a boil, then reduce heat to a simmer. Cook for 15-20 minutes, or until the chicken is cooked through.

Remove the chicken from the pot and let it cool slightly. Using two forks, shred the chicken. Discard the cooking liquid, onion, and garlic. In a bowl, toss the shredded chicken with cumin, Mexican oregano, salt, and pepper.

Pour about 1-2 inches of oil into a large, heavy-bottomed skillet. Heat over medium-high heat until the oil reaches 350-375°F (175-190°C).

While the oil heats, warm the tortillas to make them pliable. Wrap a stack in damp paper towels and microwave for 30-45 seconds. Place about 2 tablespoons of the chicken filling in the center of a warm tortilla and roll it tightly.

Carefully place 3-4 rolled tacos seam-side down into the hot oil. Fry for about 2-3 minutes on each side, until they are deep golden brown and crispy all over. Be sure not to overcrowd the pan.

Using tongs, remove the tacos from the oil and let them drain on a wire rack. Serve immediately with shredded lettuce, crema, cotija cheese, and your favorite salsa.