Bread – How to put a hot dog in a bun without splitting the bun

I'd say your problem is that the yeast has exhausted the food in the dough before you even get the leftover dough in the fridge. When you pull it out the next morning, the yeast does nothing or very little because there are no more sugars left for it to eat. And not only will this prevent rising, it will also limit browning because it's at least partly the sugars that brown in the oven.

You could work some more flour (and water, to maintain hydration) into your dough as yeast food--maybe even a little sugar if you want--and then rise/form/rise and bake. But at that point you're doing what @tmow suggests--just using your pizza dough as a starter for new bread.

Butterflying a hot dog (or any similar sausage) has two effects. First, as the moisture inside the hot dog expands during cooking, causing the casings to frequently burst due to the pressure that builds up. When you butterfly a hot dog the this is prevented. Such blistering does not 'harm' the hot dog but are somewhat 'unsightly', so I would call this an aesthetic effect. The second effect is that it allows the hot dog to cook through more evenly. Particularly for those who prefer their hot dog well done butterflying can get the insides cook through without charring the outside.

[Edit: found something interesting to add...] If you want to impress your friend with something different, try spiral cutting your hot dog.