Heat the oven to 375°F. Lightly grease a 12-cup muffin pan.
In a large bowl, combine the ground chicken, breadcrumbs, egg, onion, bell pepper, Cajun seasoning, Parmesan, milk, and 2 tablespoons of the ketchup. Mix gently until combined, taking care not to compact the mixture.
Divide half of the chicken mixture among 8 muffin cups. Add a small pinch of mozzarella to the center of each portion, then cover with the remaining chicken mixture. Shape each one into a rounded muffin and seal the cheese inside. Spoon the remaining ketchup over the tops.
Bake for 20 to 25 minutes, until the muffins are cooked through and reach 165°F in the center. Let them rest in the pan for 5 minutes before removing.
While the muffins bake, bring the water and salt to a boil in a medium saucepan. Slowly whisk in the grits. Reduce the heat to low and cook according to the package directions, stirring often, until thick and tender.
Stir the cheddar and butter into the cooked grits until the cheese melts. Spoon the grits into bowls and serve the chicken muffins on top or alongside.