Heat your Blackstone griddle or flat top grill to medium (about 350℉).
Score the hot dogs by making diagonal cuts about halfway through the hot dog (don't cut all the way through). Turn the hot dog and make diagonal cuts in the opposite direction to create a crosshatch pattern.
Place the hot dogs on the heated griddle and cook until done, rotating every so often so they cook evenly.
While the hot dogs are cooking, open the top-sliced hot dog buns and stuff each one generously with shredded cheddar jack cheese, filling the entire pocket.
Place a cooked hot dog in the center of each cheese-filled bun, pressing it down into the cheese. Add more shredded cheese on top of each hot dog.
Place the hot dogs in the buns cheese-side down on the griddle (upside down). Brush the outside of each bun generously with melted butter. Cook until the bottom (cheese side) is golden brown and crispy, about 3-4 minutes.
Carefully flip the buns over and continue cooking until the cheese is fully melted and the bun is toasted on both sides, about 2-3 more minutes.
Remove the grilled cheese hot dogs from the griddle and place on serving plates.
Top each grilled cheese hot dog with your favorite toppings like mustard, diced onions, hot dog chili, and/or jalapeños. Serve immediately.