Both my options are in 'A Day In The Death'.
First option - Owen is having a very bad week. First, he died and then he was resurrected as an undying and yet unhealing zombie. He can't sleep, he can't eat, he can't have sex. Which given that Owen is a hedonist who does all of the above to excess in order to cope with being Owen would be terrible before you get into the greater philosophical problems of being undead.
Owen, understandably, is upset by this, but because of who he is, the way he explains this to Tosh is by breaking two of his fingers, which he knows will never heal. And yes, he does it in the heat of an argument, but he knows exactly what he's doing. And that scene is a lot of Owen's character distilled into one scene.
But what the meme is asking for is favourite moment, not most intrinsically Owen moment. And that would be the end shot of 'A Day In The Death', where Owen is holding Maggie's hand, and even knowing what he knows, they both decide that this is better than nothing. And it's a very hopeful moment in a show that doesn't really have them, with a character who doesn't really have them and it's the other part of Owen coming through, the part that does keep trying to save the world, despite everything and it's just lovely.
Day 1 -
Your favourite Torchwood memberDay 2 -
Your favourite Torchwood seasonDay 3 -
Favourite EpisodeDay 4 -
Favourite alienDay 5 -
A scene that made you cryDay 6 -
A scene that made you smileDay 7 -
Your Favourite QuoteDay 8 -
Your favourite cast memberDay 9 -
An image that makes you smile like a gitDay 10 -
Your favourite promo pictureDay 11 -
Your Favourite CoupleDay 12 -
Favourite Couple SceneDay 13 -
A piece of music you love from the seriesDay 14 -
Something you’d like to re-write in the seriesDay 15 -
Behind the scene video that made you laugh the mostDay 16 -
Your favourite outfitDay 17 -
Your favourite Gwen momentDay 18 -
Favourite Ianto momentDay 19 -
Favourite Jack moment Day 21 - Your favourite Toshiko moment
Day 22 - Who you’d shag, cliff and marry
Day 23 - Your favourite Torchwood website
Day 24 - A Torchwood geek moment you’ve had
Day 25 - Your favourite Torchwood fan-fiction
Day 26 - A piece of fan-art
Day 27 - A fan-video
Day 28 - Your favourite location or set
Day 29 - Where you think the series should continue to
Day 30 - Best Torchwood on Doctor Who moment