Dumping the effluent onto the deck would be out of the question, in my view. At the roof, instead of a radial fan in a housing blowing air down the roof to the deck, use an "upblast" fan housing with compatible fan. Broan may still make such a unit (NuTone did) or search on the web. Even Greenheck may make one small enough. The upblast housing roof interface will have to be the correct pitch for the roof, and, if snow builds up very high in that location, high enough to keep the top out of the snow. kas

Thanks everyone. It's an open-air patio, not enclosed. The overhang does have vented soffit :( Also, I've never had a vent hood before. The smells of my cooking are not bad but they do linger, thus the hood. Boxer, what do you mean snake the tube? With a chimney style hood there can be no snaking until you get past the drywall. My problem is that there is a joist in the way. Actually, a ceiling joist, not a roof joist. We are trying to avoid cutting joists. Is there some way of snaking that I'm not aware of that will make a chimney style hood work? I know I can get an undercab hood or an insert and angle it within the cabinetry to avoid the hood. The chimney style would be better for the aesthetics of the plan and let more light into the adjoining room. I'd like to ask him to move the troublesome ceiling joist and reinforce to in order to fit a roof-vent chimney mount style, but that's where the roof slopes to its cramped little end and will be a &(&*^% to do.. I guess we'll see. He does like the chimney style as well, so I'll let him decide if he wants to deal with the joist or not.

For the wall vent option, also consider the proximity of operable windows. Codes vary by locality, so you should probably call you local building inspector for clarification, but in my city in Northern California the vent must be >=3' from any operable windows. Folks above are correct that 90s in ducts result in pressure drop, but you're talking about a short run with a single bend, and you could always run a slightly larger duct if you were concerned about it, so I don't think the pressure drop from the single turn is a significant consideration because it's gonna work well either way. Primary considerations are: cost to install, how it looks, if smoke will find it's way back into house or otherwise annoy outside.

It'll be fine. Don't worry too much about it. It might look funny to you the first and second week, but afterwards you won't even notice. But what you will notice is the much better venting and better air quality. Make sure you buy a good vent hood that has adequate power and overhang. Since you can't really do anything about overhang on the sides, at least try to find a hood that is extra deep. Given you have such tall ceilings, consider mounting the hood above head height. That really helps with sight lines. And it works well in combination with having an extra deep hood. Something like 27" depth would be awesome. Tell your contractor to install big diameter ducts. Something on the order of 8" or 10". And get a hood with baffles instead of mesh filters. For more helpful recommendations, read the FAQ on this site or refer to postings by @kaseki. They are all invaluably useful and insightful.