Make the stock:

1. Cut the legs from the chickens. Pull off the skin that covers the breasts, then remove the breasts from the carcasses. Cover and refrigerate or freeze the breasts for another use. Cover and refrigerate the legs to make the ramen.
2. Bring a medium pot of water to a boil. Add the chicken carcasses one at a time and cook for 30 seconds to blanch and clean the chicken. Transfer the carcass to a very large pot, arranging in a single layer in the center. Strain any scum off the top of the water and bring back to a boil. Add the chicken feet and cook for 60 seconds. Transfer the chicken feet to the pot with the carcasses, arranging them around the carcasses.
3. Place the onion quarters down the center on top of the carcasses. Add the water. Bring to a simmer uncovered over medium-high heat, watching closely and reducing heat to maintain a simmer and skimming off any scum as soon as it forms at the top of the stock. Do not let it come to a boil. Continue to simmer, skimming the stock as needed and maintaining the heat to ensure it simmers and does not boil. Try not to agitate the stock to prevent it from turning cloudy. The stock will start out with a greyish color but will turn golden as it simmers. When you start to notice the color of the stock changing, after about 1 hour, turn the heat to low and cover the pot with the lid slightly ajar. Simmer until the stock turns golden yellow, about 3 more hours.
4. Meanwhile, turn a gas burner onto medium to medium-high. Hold on to the dark green part of the leek with your hand or tongs with the white part of the leek over the flame and cook, turning occasionally, until the white outer layer is charred all over. Alternatively, this can be done under the broiler. Cut off the dark green part of the leek and store the charred leek airtight until cooled.
5. Peel off the outer charred layer from the leek and add to the stock. Add the cabbage, mushrooms, katsuobushi, if using, Parmesan rind, ginger, and rock sugar. Simmer the stock, covered with the lid slightly ajar, until the stock is golden and aromatic, 2 to 4 more hours.

Make the chicken adobo ramen:

1. Place the chicken legs on a plate. In a small bowl, stir together salt and sugar. Sprinkle all over the chicken legs. Cover and refrigerate for 30 minutes. Rinse the chicken and pat dry.
2. In a small skillet, heat chicken fat over medium heat. Add the 2 tablespoons of finely chopped onion and cook, stirring often, until onion is deep golden brown, about 5 minutes. Transfer about 2 tablespoons of the chicken fat with onion into a large heavy skillet. Heat over medium-high until shimmering. Add cured chicken legs, skin-side down, and cook until golden brown, about 5 minutes. Turn chicken over and add sliced onion to the skillet. Cook, stirring often, until translucent, 3 to 5 minutes. Add garlic and stir until fragrant, 1 to 2 minutes. Stir in ginger.
3. Add spices to the skillet, along with bay leaves and pepper. Cook, stirring occasionally, until fragrant, about 1 minute. Add brown sugar and stir until dissolved, about 1 minute. Add chicken broth, vinegar, and soy sauce. Stir in coconut milk, if using. Cover with a tight-fitting lid, reduce heat to medium-low to low, and cook, stirring occasionally, until chicken is meltingly tender, about 2 hours.
4. Meanwhile, cut dark green scallions lengthwise in half, then open up on a cutting board. Thinly slice the scallion lengthwise to create long thin strands. Place in a medium bowl of ice water until ready to serve.
5. Bring a medium pot of water to a boil. Add spinach to the boiling water and cook just until wilted, about 1 minute. Using a hand-held strainer, transfer spinach to a work surface. Let spinach cool, then roll into a log shape about 6 inches long. Cut crosswise into 4 equal portions and reserve for the topping. Fill pot with fresh water and bring to a boil for noodles.
6. Remove chicken legs from the skillet. If needed, whisk a few tablespoons of stock into skillet to loosen the juices and browned bits from the bottom of the skillet. Carefully lift the thigh meat from the bone in one piece and set aside. Shred the cartilage and scraps from the leg and reserve for the garnish.
7. Add the noodles to the boiling water and cook until tender but still firm to the bite, about 1 minute. Drain the noodles. Drain the scallions and pat dry.
8. Divide the adobo sauce in the skillet between 4 deep bowls. Add the chintan stock to the bowls, stirring to blend with the adobo. Add the noodles to the broth, picking up and folding over 2 or 3 times (pick up, fold forward, then backward.)
9. Top each bowl with some of the grilled corn and one chicken thigh. Garnish with spinach. Divide the scallions into 4 portions and roll each to form a nest. Place on top of the chicken. Garnish with the cartilage and chile threads, if using. Drizzle with peppercorn vinegar, if using.