I agree with your interpretation of "thus thrown proved buttons". "This side the oesophagus" means "this side of the oesophagus". This omission of "of" is old-fashioned or literary/poetic.
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wholegrainAlso, I found something peculiar in this passage. He says "this side" as if he was indicating a side with his finger instead of "the side of the oesophagus" which I think would make more sense, at least to me.