"Bring home the bacon" = Going out and working to make a salary. "Bring home the bacon" refers to spending your salary on food for the household... It is similar to the term "breadwinner".
As for the sentence you have given ['I can bring home the bacon mm mm mm mm fry it up in a pan mm mm mm mm.'], I think it's a little bit of a joke. It's talking about "literally" bringing home the bacon and cooking it up. At the same time, it's referring to making money and then coming home to "enjoy it" / "feel satisfied". It's the way for E to say: "Hey! I can make money too...".
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'And that, as such , they ' re naturally going to want what any human being faced with two IRREVOCABLY conflicting sets of responsibilities is going to want . Meaning that what they ' re really going to want is some way out of these responsibilities.'
The sentence means that when someone has two responsibilities that conflict with one another [contradict each other], they often want to escape both.
To give an example. A man is a policeman and also a father. A robbery happens at a store in town and the policeman is assigned with solving the case [finding out who did it]. He discovers that his son is actually the robber! Now he has two CONFLICTING SETS OF RESPONSIBILITIES:
1) A responsibility as a policeman to report the crime and resolve the case;
2) A responsibility as a father to protect his son.
The two responsibilities are conflicting and the father is faced with a very difficult decision. At such times, most people would wish to have neither responsibility!! [That's what your sentence is saying.]
With regard to the context of the text, it is suggesting that women find it difficult to have both household and work responsibilities, because these conflict with each other.
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I hope this helps!
Good luck and take care!