While waiting for the meat to cool, return the stock and fat to a boil. Reduce to approximately 7 cups.
Bone the chicken and rabbit. Chop the meat into 1 inch cubes. Dice the giblets and reserve all meats in a bowl
Render the finely chopped side-meat in the Dutch oven over medium heat.
Add the onion, celery, and carrots. Cook until tender, about 5 minutes, stirring often. Add the garlic, thyme, bay leaf, red pepper flakes, and smoked paprika. Toss to combine. Stir in the tomatoes and all their juices and bring to a boil. Reduce heat and simmer for 20 minutes, stirring as needed.
Add the lima beans, potatoes, and corn. Return to a boil and stir in the meats and giblets with the stock and fat. Bring to a boil again, and then simmer until the stew thickens--about 1 hour. Stir often to prevent sticking.
As you proceed through the steps above, season to taste with salt and freshly ground black pepper. Want more heat? Add some red pepper flakes or paprika. Let it simmer a few minutes and taste again before seasoning further.