When someone says "the something" and i don't understand that something they said I say "the what" in the idea of letting them know I didn't understand what they said.
How horribly I butcher English by using that construction?
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You aren't butchering English by saying "The what?" as a question. That's entirely idiomatic. In fact, if someone seems to have said a name that you didn't catch, you can also say "The who?" (or just "Who?", if the original sentence didn't have the definite article "the" beforehand).
In both cases, the idea is that you give just enough context, usually with "the", to let them know what part of the sentence you're asking about. Sometimes you need to include more. If someone says "I'm going to poke the ??? with the long pole", you might say "Poke the what?" so they know you're not asking them to repeat "the long pole".