In a shallow dish, coat the trout with the cornmeal. Heat a large skillet over medium heat and add 3 tablespoons of the butter. When the butter begins to melt, place the trout, skin-side down, and cook over medium heat until the skin turns brown and crisp, about 5 minutes. Gently turn the trout over and continue to cook another 3 minutes. Season with salt and pepper, to taste. Reserve the pan juices.

Melt the remaining butter in a medium saucepan over medium heat. Add the carrot and onion and cook until the onion is just translucent, about 10 minutes. Add the celery, black-eyed peas, bay leaf, and thyme. Cook for 5 more minutes. Add just enough water to cover the mixture and lower the heat. Cook until the black-eyed peas are tender, about 1 hour and 30 minutes. Add the strips of country ham. Season with salt and pepper, to taste.

Remove thyme sprigs. Place the warm black-eyed peas in the center of a plate. Arrange the trout on top. Drizzle with the pan juices.