Cut the mozzarella and cheddar cheese sticks in half both vertically and horizontally to create 24 pieces of each.
Join one of each cheese type together to create a single dual-colored cheese stick.
Create a crumbing (breading) station with three shallow dishes. In the first add the flour, in the second whisk together the eggs and milk, in the third add the breadcrumbs and melted butter and stir until well combined.
Dip the cheese in the flour until lightly dusted all over.
Dip the cheese in the egg mixture then directly into the breadcrumbs. Place on a plate. When all 24 are done chill for 20 minutes.
Removed the chilled cheese then repeat the egg mixture, crumbing, and chilling process two more times.
Preheat a grill with a hot zone and a cold zone. If using a round grill, place all the coals in the center with the edges clear. The mozzarella sticks will be cooked in the hot air not directly over the direct heat of the coals.
When the grill is preheated to 350 degrees F (180 degrees F), place the mozzarella sticks on the grill away from the direct heat.
Cook for a total of 15 minutes turning every five minutes until golden brown on the outside and the cheese is soft and gooey on the inside. Serve immediately.