Weird Aside

During what had previously been a perfectly ordinary conversation, one character nonchalantly inserts something completely bizarre, and often disturbing, which often derails the conversation. They may do this knowingly or innocently, not realizing how strange what they are saying is. This is often used for humorous purposes (even by the very character, if knowingly) or to add emphasis to how screwed up a life the particular character had, that they think something like that would be normal or conversation-appropriate.

Here's an example:

Old Lady: "Oh, this recipe? Well, when I fought the militias in Qurac, I met this woman who taught it to me. Very sweet lady, even though her husband made that Deal With the Devil. Anyways, first you take two eggs, separate the whites and the yolks... Are you listening?"

Similar to Bread Eggs Milk Squick, only it's in a conversation and it's actually Bread, Eggs, Milk, Beat... wait, what? It can also be Bread Eggs Milk Squick depending on the content of the aside.

Film

 * Mortimer Brewster's aunts in Arsenic and Old Lace tended to speak much like this.

Live Action TV

 * Phoebe Buffay, from Friends, often begins a story with "When I was living on the streets..." or "When my mother killed herself..." - The titular friends take it in stride (after all, they already know about Phoebe's mother and life story), new acquaintances do not.
 * Also, when Phoebe was a Surrogate Mother she liked to freak people out by talking about having her brother's baby.
 * Chandler also had a penchant for this, especially regarding his father - though he was quite aware of how weird it was.
 * "The word you're looking for is, 'anyway...'"
 * Sue Sylvester in Glee frequently does this.
 * The Janitor in Scrubs does this sometimes.
 * The news segments on Top Gear often go off-topic for one reason or another, but no one derails them faster than James May.

Literature

 * Some of the wizards in the Discworld series have a tendency to make awkward pauses in the conversation even more awkward by filling them in with random trivia. In Reaper Man, the Senior Wrangler is said to be able to "do to a conversation what it usually takes thick treacle to do to the gears of a precision stopwatch."

Web Animation
"Sister: I thought about having a kid once.
 * The character Sister from Red vs. Blue is known for this. She inserts all kinds of weird stuff into conversations that make people go "Wait, what?" as if it were normal.

Tucker: Oh really? It's a lot of work.

Sister: Yeah, it seems like it would be hard. But I thought, you know, who wants to be known as the girl who's had seven abortions?

Tucker: Heh, yeah... wait, what?"

Western Animation
"Spike: You know, Pinkie, these two ponies have a bit of a grudge match they're trying to settle, trying to prove who's the most athletic.
 * Pinkie Pie from My Little Pony Friendship Is Magic is seen doing this while doing the commentary for the Running of the Leaves in "Fall Weather Friends":

Pinkie Pie: Yes! And grudge rhymes with fudge!

Spike: Yes it... does... what?

Pinkie Pie: And I like fudge! But if I eat too much fudge, I get a pudge, and then I can't budge.

Spike: So...no fudge?

Pinkie Pie: No thanks! I had a big breakfast.

[...]

Spike: Looks like Rainbow's doing her best to catch up!

Pinkie Pie: I'm not sure how ketchup is going to help her in this contest. Now in a hot dog eating contest, it can make them doggies nice and slippery, but personally I prefer mustard. How about you, Spike?

Spike: Uh...I like pickles?"

Real Life

 * Alfred Hitchcock did this in real life, especially as he was getting out of an elevator. Just as he was getting off at a floor, he'd turn to a friend and say, "So there I was with the dead body. Looking back I shouldn't have touched the knife, but, with such an amount of blood before me, I acted with poor judgement." As the elevator door shut, the other passengers would wonder what the hell had happened.
 * Ross Noble's entire routine is this. He begins practically every show by observing something strange an audience member is wearing or doing, and goes on to make a two-hour show out of it. A single show could cover Hare Krishnas, sailors, emus, and DJ Aslan before the night is out.