Hmm... "One soup pot's handle is broken" doesn't sound quite right to me: that suggests that you have a few different soup pots in front of you, and one of them has a broken handle. It also implies that the broken one is the only handle that soup pot has. The other correct solution - "One of the saucepan's handles is broken" - comes much closer to the meaning of the Russian sentence, I think. In everyday speech, you might also hear a noun used as an adjective in this situation ("One of the pot handles is broken").
I have two saucpans that each have two handles. We always refer to them as saucpans as we also have frying pans, milk pans, a skillet and a steamer the base of which has two handle of equal size as does each of the two steam sections on top. All of them are heated on the hob.