As a home brewer I often have this problem when malt caramelizes at the bottom of the pot. I use Sodium hydroxide (NaOH), also known as lye and caustic soda. depending on the strength of the product you can get your hands on, for the one I can get, a 3% dilution in water is enough, but you should read the indications of the manufacturer anyway. simply fill with solution until it covers all of the stuff you want to remove. let it sit for a day or so and the stuff will be floating by the time you see it again. No damage to your pots, no need to scrub anything, despite what you might think it is safe to consume trace amounts of this stuff at the recommended dilution(you might want to give it a good rinse anyway). this is what food processing plants and breweries use to clean their equipment after all.
Sodium hydroxide (NaOH) should only be used on food grade stainless steel, if your pot is made from aluminum you CANNOT use this, as Sodium hydroxide reacts with aluminum and pretty much dissolves it.
If your are unsure if your pot is made of steel, a fridge magnet can be used to test it, if it doesn't stick to it: it's not food grade stainless steel.
Sodium hydroxide (NaOH) is a dangerous chemical that should be used with care, always wear eye protection and gloves.