What's an effective way to remove this stubborn burnt-on oil from a saucepan?
If your pan is truly stainless steel, you can get burned on brown and sticky oil off the sides with oven cleaner (US brand "EZ-off" or other brand). This product is definitely not recommended for other metals such as copper, aluminum, etc.. But true stainless steel, I have used it no prob. If you are in doubt, you can test a small amount out on an inconspicuous tiny spot at first.