Heat your gas grill for a two-level fire with one side set to medium-low and the other cranked up to high. Cook the hot dogs for about 1 1/2 minutes on the cooler side, then move them to the hotter side and turn them regularly until they're cooked to your liking. I like them with a few char spots!
Add blue food coloring one drop at a time to tint the mayonnaise blue. Transfer the blue mayonnaise into a piping bag, then snip a small opening at the tip. Transfer the ketchup into your other piping bag and snip a small opening at the tip as well.
Toast the brioche buns however you like. For the grill, toast them cut-side down on the grates for 1 minute.
To make the white stars, press a small star-shaped cookie cutter through the slice of white cheddar cheese. Repeat until you have enough stars.
Now you're ready to assemble! Line each toasted bun with a lettuce leaf, then lay a few pickled cucumber slices on top. Place the cooked hot dog in the center, then pipe clean lines of ketchup and mayonnaise down its length. Finish the 4th of July hot dogs with cheese stars on top, and you're done!