Preheat a grill to medium-high.
Heat a large skillet over medium to medium-high. Add the butter. When it melts, add the onions and garlic; season with salt and pepper. Cook, stirring often, until the onions are very soft and deep golden brown, about 20 minutes. Add the hot sauce and reduce the heat to low, adding a splash of water if the onions get too dry.
Grill the hot dogs, turning occasionally, until dogs are heated through and casings are crispy, about 4 minutes.
Place the dogs in rolls; top with the onions, carrots, celery, relish and cheese.