Heat the butter in a large stockpot over medium heat. When it melts, add the onion, carrot, celery, bell pepper, jalapeno and 1/2 teaspoon each salt and pepper. Cook, stirring, until the vegetables are soft and sweet, about 20 minutes. Add the garlic and tomatoes and cook until thick and jammy, about 10 minutes.
Meanwhile, preheat a grill or a stovetop grill pan over medium-high heat. Blot dry the chicken legs and pork and season with salt and pepper.
Grill the chicken and pork until wellmarked on both sides and partially cooked through, about 10 minutes. Cut the pork into 1-inch cubes.
Add the thyme, paprika, cinnamon stick, 1 teaspoon salt and 1 teaspoon pepper to the stockpot. Add both meats, the stock, 4 cups water and the bay leaves. Bring to a simmer and cook, partially covered, stirring once in a while, until the pork is very tender when poked with a fork, about 1 hour, 30 minutes.
Remove the chicken legs, pick the meat from the bones, chop it into large pieces and return it to the stew.
Add the hominy and simmer the stew another 30 minutes to 1 hour, or until everything is really tender and the meat is falling apart. Mash the pork against the side of the pot to break it up. (Don't worry if the chicken looks like it's almost dissolving: That's typical of booya.) Add the corn and cook 5 more minutes.
Remove the bay leaves and cinnamon stick. Taste to check the seasoning and add more salt, if needed. Serve with the chopped cilantro.