Prepare a grill fire to 350° using pecan or hickory for smoke flavor.
Make a pocket in the center of the chicken breasts by slicing through the center from one end, making a 2- 3 inch slit, not cutting all the way through.
In a bowl, combine, ham, 1 cup cheddar cheese, 1/4 cup diced onion, 1/4 cup diced bell peppers, 1/2 teaspoon salt and 1/2 teaspoon pepper.
Stuff filling into chicken breasts.
Place on a baking sheet and brush or spray with olive oil. Sprinkle with 1 teaspoon kosher salt and Everything Bagel Seasonings.
Cook chicken over indirect fire with the grill closed for about 25 minutes. Check internal temperature using a digital thermometer in the top of the chicken breast. Cook until chicken reaches 165°.
Remove to a platter and rest five minutes.
Prepare the cheese sauce while the chicken is cooking. Melt butter in a skillet and cook 1/2 cup diced onion, 1/4 cup diced bell peppers and the minced garlic clove over medium heat, stirring often until vegetables are softened.
Add flour and whisk to combine. Continue to stir and cook about 2- 3 minutes. Whisk in milk.
Add cream cheese and whisk until fully melted. Lower to simmer and mix in 2 cups shredded cheddar until fully combined and sauce is smooth. Keep warm until ready to serve.
Drizzle about a tablespoon of sauce over each chicken breast and move to a cutting board. Slice into thin slices, cutting across the wide part of the chicken, so that the filling stays inside the pocket.
Serve sliced chicken over rice with additional cheese sauce. Garnish with parsley if desired.