That was the traditional rule, but in modern English, "whom" is a particularly marked usage anywhere except as the immediate object of a pronoun. "With whom am I speaking?" sounds formal but fine; "Who am I speaking with?" (or "...to?") is also fine. The form given in the title here is at best overly formal and stilted. What makes it weird is that it mixes the formal not-after-a-preposition "whom" with a sentence-ending "with", mixing register in a very confusing way.

It's like wearing a bow tie and tails with loud bermuda shorts: both are perfectly acceptable public attire and you might see either on a busy urban street of an evening, but it's jarring to see them together.

Who is a subjective pronoun. Whom is an objective pronoun.

First hit off the Google: http://web.ku.edu/~edit/whom.html

Doesn't matter really, just made my brain twitch when I read it.

The reason it "sounds weird" is that it's attempting to straddle two dialects. The use of "whom" is a signal for classical (Latinate) grammar, which would demand the Latinate prepositional word order, while the use of separable-prefix Germanic word order with "with" at the end is a signal to use "who" because dangling prepositions are associated with the less-Latinate trend towards discarding accusative "whom".

These modes normally don't get mixed, hence the weird sound of the title of this post.

I think it sounds odd because it's disconnected from its preposition. In fact, I'm not sure I've ever heard a sentence starting with "whom."

It wasn't a troll, but you're correct. The word order apparently caused my brain to spasm.