For the basil chimichurri: Add the basil, oil, parsley, onion, vinegar and salt to a bowl and stir with a fork to combine. The chimichurri is best served immediately, while the herbs are still fresh, but it can be stored, covered, in the refrigerator for up to 1 day.
For the chicken: Heat the olive oil in a large cast-iron skillet or Dutch oven braiser over medium-high heat. Sprinkle the chicken with the salt and pepper on both sides, then place the chicken pieces skin-side down in the skillet. Fill in the skillet with the potatoes. You want the pan really full so that the surface area is completely covered with chicken and potatoes. Cover with a lid and cook until the chicken has released its juices and everything is golden, 12 to 15 minutes.
Flip the chicken, then cover again and cook until the thickest part of the chicken reaches 165 degrees F, another 8 to 10 minutes. Remove from the heat, then add dollops of chimichurri on top and serve immediately.
