Sprinkle the chicken generously all over with salt, pepper and cumin. Let rest for 30 minutes at room temperature.
Add the olive oil to a soup pot on medium-high heat. When the oil begins to swirl, add the chicken skin-side down. Cook, undisturbed, until a peek beneath reveals a golden crust, about 5 minutes. Remove the chicken to a plate and continue. (The goal here is to just get flavor and fat from the chicken, not cook it.)
Add the jalapenos, onions and a pinch of salt to the pot. Cook, moving the veggies around until they become golden in spots, about 4 minutes, then add the sofrito and soup base and bring to a bubble. Remove the skin from the chicken and discard. Add the chicken back to the pot and simmer, stirring occasionally, for 5 minutes. Add 4 cups water to the pot and bring to a bubble again. Lower to a simmer and let cook for 15 minutes.
Add the egg noodles and cook until tender, about 3 minutes. Just before serving add the cilantro and stir. Divide into 4 bowls, giving each a piece of chicken.