Prepare a grill for medium-high heat. Put the hot dogs on the direct heat of the grill and cook until desired doneness.
In a pan on medium heat, add the olive oil, breadcrumbs, a pinch of salt and few grinds of pepper. Toast, stirring, until the breadcrumbs soak up the oil and become golden, a few minutes. Toss in the parsley and basil off heat.
Open each bun and stuff with equal amounts of mozzarella. Wrap in aluminum foil and place over the indirect heat of the grill (or in a warm oven, about 300 degrees F). Remove when a check inside reveals melted mozzarella. Spray the tops of each lightly with butter spray, sprinkle the breadcrumbs over the top of the buns, return to the grill or oven and cook until the tops are a bit crusty and stiff, a few more minutes.
Put the hot dogs over the melted mozzarella in the buns. Top with pizza sauce and grated Parmesan.