Serious buildup outside of cast iron skillet

The inside looks absolutely fine, so assuming you will protect that if you do any actual work on the outside I see a couple of routes you can take...

1. You can ignore it & see if it will reach a new equilibrium with your own cooking equipment & technique.

2. You can clean it off & quite quickly return the outside to a lesser seasoned state - as you're not cooking on the outside the seasoning is far less important, & as you've already discovered, the outsides of pans can get a lot more build-up over time than the insides*.
   You could attack it with sandpaper, or a drill & wire brush attachment, though you need to beware of the mess that can make if you do it indoors... or you could take it to somewhere that could shot-blast it for you. This is an industrial cleaning process, depending on abrasive type can be extremely vicious, but it's rapid & they could clean the outside back to shiny iron in 2 minutes.

You can then re-season the outside.

*You should have seen my old wok before my partner threw it away. I eventually forgave her ;)

I inherited old pans with thick carbon build up. I built a fire in my fire pit and placed the pans above the coals. After all these were pans my parents used on campfires. I kept an eye on them until the carbon started flaking off. I would not place them on a roaring fire you could run the risk of cracking your pans. Now this is something I would not recommend if you have valuable cast iron. I have a friend whose grandmother would season her skillets in an earthen oven completely surrounded in fire. She did this for decades as it was her custom to give cast iron as gifts. She never once cracked a pan. She gave me a griddle to make tortillas on that I’m still using 20 years later.

Placing the pan in the oven during a self cleaning oven cycle will make the peeling layers turn to ash, and they can be easily scoured off using a steel wool scrubber. (Tried this from experience with a pan very similar to yours)

Unfortunately, this will require re-seasoning the entire pan.

I am always looking for things to break out my dremel....iT would work perfectly on a cast iron pan that has build up...In fact you could used several different attchments to really take off the black bumpy stuff....I would have a ball if I had a cast iron skillet that was old and needed this I have it looking new in no time at all and you could too !! grab your dremel and get to work !