Heat 3 tablespoons of vegetable oil in a large skillet over medium heat.
Add 1.2 kg of cut chicken to the skillet.
Season the chicken with 1 1/2 tablespoons of oregano, 2 minced garlic cloves, and salt and pepper to taste.
Cook and stir the chicken until it is lightly browned.
Add 1 sliced onion to the skillet, then cover and reduce heat to low.
In a blender, combine 10 tomatillos, 1/2 bunch of cilantro, 2 jalapeno peppers with seeds removed, 2 whole garlic cloves, juice of 1 lemon, and salt to taste.
Pulse the blender to achieve a chunky consistency.
Pour half of the tomatillo mixture over the chicken in the skillet.
Continue simmering the chicken for 35 to 40 minutes, until tender.
Serve the chicken with the remaining sauce and top with 1/2 cup of shredded Monterey Jack cheese.