Just Some Chickadee Analysis
In regards to @darlingditzy1966's post regarding my Chickadee art:
Absolutely, I think that Chickadee would see the Penguin as a role model! I believe she'd look up to him a lot, because he's very refined in what he does, and even though he presents himself in such a way, putting on a show in the same way a lot of rish folks would, with his top hat, cigarette holder, monocle, outfit, etc., his crimes really get quite terrible when you think about them too hard. Which, don't, because that defeats the purpose of the show, but the hinting toward it is there, it's coded within the moment.
Then thinking about Chickadee, what makes her character such a contrast? She appears sweet and fun on the outside, but then has an almost morbid fascination with carrying out the crimes that she gets taken along on. It's the same concept! Appearing one way, and acting another. The same way Penguin appears distinguished, but is in actuality one of Gotham's arch criminals, to the point where he has a reserved cell in the penitentiary, LOL (which is something we find out in Chickadee's episode - "The Penguin's Nest/The Bird's Last Jest", Season 2, for anyone curious; it's one I recommend!).
And as Ditzy says in the post, Chickadee is very keen on using the umbrella gun. We're getting into headcanon territory, now, but here's what I think, and forgive me for it's been a while since I've seen the episode (although I think I'll watch it again later, now), but I believe getting to use the umbrella gun is something that she always dreamed of, in addition to working for the Penguin.
Getting into some character analysis here, but before Chickadee started working for Penguin, she was still a criminal, but was a nortorious bootlegger of untaxed cigars. Now, we've seen how excited she gets during grand schemes, when things get particularly tense or violent, and how it almost energises her, so does it not seem a little odd that as her own criminal, she wasn't in a similar business? Something more exciting toward her tastes, I mean. Obviously in-show, it's just a little joke to link them, but suspending our disbliefs for the sake of cool connections, could it be that she was in the cigar business on purpose, as a way to try and get connections to the Penguin?
Gotham Penitentiary has a seperate wing for the women, so even if she was rather famous for her crimes and was arreasted, there'd be no way she could by chance come across him on the inside, no - she needed something that established she was on his side of the law, and would also get her linked to him, and what better way than cigars & cigarettes, something he's always smoking?
It would make sense for her to have figured this, too, because Chickadee is pretty smart, to her credit, but in a very specific way. Impulsively, and in a very "if I do (x) thing, (y) will happen" fashion. Her type of intelligence works on that basis, rather than something to like, say, Riddler's standards. She couldn't orchestrate an entire scheme or come up with a really intricate/important conclusion, but she can think on the fly. We see her do as such when Chief O'Hara shows up to the jail cells, and she immediately thinks to trip him up and hold the gun to him in order to get Batman and Robin to stop fighting Penguin and the other goons. It's very cause and effect, and she attempts a similar trick again later with Aunt Harriet.
This established, we'll now talk about it being her dream to use the trick umbrella. She was already plenty excited to be handling a regular gun and moreso when pointed at O'Hara, as seen by how she uses her cutesy teasing to bring the dynamic duo's attention to the fact. I believe she thought to do this because she wanted to prove initiative with guns, in order to be allowed to use one of his trick umbrellas later on. This is because once her little scheme works, Penguin praises her with "Good work, Chickadee!" in a very pleased manner (I believe he laughs about it too, but I may be wrong), showing that he now sees she has good initiative and can be trusted to help out in such circumstances.
And what is she allowed to wield next time a firearm needs to be used? The trick umbrella gun. She proved her usefulness earlier on, and by the immense joy on her face as everything goes on, it's clear she finally got what she wanted, that dream this was all building up to, being that asset. And why would she think this works in the first place? Because of the cause and effect thinking I dicussed earlier.
Further, she even tries the same trick on Harriet as I said, under incredibly similar circumstances to the jail fight, Penguin and his goons losing a fight against Batman and Robin. It worked before, she was congratulated for it and got what she wanted out of the ordeal, so why wouldn't she try again? After all, it's the way she thinks!