I'm surprised so many people think it's perfectly acceptable to just ignore the email.
If a poster came here and said "My best friend just emailed me asking if I might be able to pick up her mail while she's on vacation next week. I really don't want to, but don't know how to say it. (And incidentally, my friend has catsit for me while I'm on vacation in the past.) Can I just ignore the email and pretend it never happened?" would you tell this hypothetical poster that they were perfectly polite to do so?
More like:
"A woman in one of my social groups asked me if she could borrow my mink coat . They've been around me enough to know that I don't loan this particular coat out. She has let me borrow a wool pullover once. I've side-stepped the question a number of times in her presence and am always coy when people talk about how great it would look with their favorite dress. I feel shocked that she would ask, and very put on the spot. Do I have to respond?"
I would say Nope. An email - like an invitation - is not a summons.
That's a great analogy! I see it the same way.
I do too. Sometimes pretending not to see someone's faux pas is the kinder course of action. It's not as though the OP asked to use a tourniquet and was ignored.
How is it kind if a person does not know they commited this faux pas 
I am afraid I don't understand this reasoning.
I agree. It's kind to ignore it if someone passes gas, or calls you by the wrong name or something. Those are things that are clearly, in our culture, things that you're not expected to do. In fact, the OP was kind enough to do just that with the first email, and assume that it wasn't delivered, and give the friend an out for not answering when she should have, and the friend is STILL answering her with rudeness.
I really don't see how the OP's request is a faux pas, at all, in any way. Just because something makes someone uncomfortable, doesn't mean it's rude. Just because Betty is uncomfortable seeing women in bikinis, doesn't mean that it's rude for Jane to wear a bikini at a place where swimsuits are appropriate.
There's just nothing rude about asking a friend (or even a stranger) for a recipe, unless you've already asked that person
and been turned down. Having seen someone give a coy answer to someone else in very different circumstances just doesn't count. If a near stranger at a party asks me to borrow $5, and I say no, and BF happens to be in the same room, does that mean BF can never ask to borrow $5? I think it's pretty ridiculous to expect all of your friends to read your mind and know whether or not you're willing to do something. It's a perfectly natural assumption to think that if you've shared recipes with someone in the past, they might be willing to share one with you.
It would have been rude if the OP had sent an email that said "Give me your recipe." Or if she'd sent a second email saying "Well, you're obviously ignoring me, over a stupid recipe! You're so selfish!" But really, if you're sooooooo incredibly sensitive about your supersecret recipes that you can't even bear the idea of sending a polite "no" to someone when asked, perhaps it's time to stop cooking for anyone but yourself and your family. If someone can't even send a simple no, by email, when given a chance to say no, to a good friend, for a small, normal favor, then honestly, she has some growing up to do. If she doesn't want to share something that's often shared (and despite insistence to the contrary, mink coats are not often shared), it's on her to learn to say no. I'm amazed, frankly, at how many people are advocating such a major snub in reaction to a simple, honest, polite request.
And it sounds to me like the friend is pretty good in general about replying to email, so I don't think the "maybe she was busy" excuse cuts it. Even if she is busy, when I get a second request on something, my reaction is usually something like "Oh crud, I forgot to answer. How rude of me," and I'll send out a quick line, at least to let the person know that I saw the email and was working on it.
I do agree that the OP should drop it at this point. Clearly, the friend is more concerned with not having the slightest second of discomfort than with treating the OP with a bit of basic human respect, and I think that shows a LOT about the friendship.
