Melt 1 tablespoon of butter in a large Dutch oven over medium-high heat. Brown the pork shoulder cubes on all sides, about 6–8 minutes, then remove and set aside.
Add the diced onion to the same pot and cook until softened and translucent, about 4 minutes.
Return the pork to the pot along with the chicken thighs, crushed tomatoes, chicken broth, bay leaves, salt, and pepper. Bring to a gentle boil, then reduce heat to low, cover, and simmer for 45 minutes, until the chicken is cooked through.
Add the diced potatoes and lima beans. Cover and continue simmering for 25–30 minutes, until the potatoes are fork-tender.
Stir in the corn, ketchup, brown sugar, Worcestershire sauce, apple cider vinegar, and hot sauce. Simmer uncovered for 15 minutes.
Remove the chicken thighs and shred the meat off the bone. Return the shredded chicken to the pot along with the pork.
Using a potato masher, mash about half of the potato chunks directly in the pot, leaving the rest whole. Simmer for another 10 minutes.
Stir in the remaining tablespoon of butter. Remove the bay leaves, taste, and adjust seasoning with additional salt, vinegar, or hot sauce as needed. Let rest for 10 minutes before serving.