What are your goals for your chickens? What do you want from them? It’s hard to recommend a breed if we don’t know why you want chickens and what you want them to do. What I suggest is that you go through Henderson’s Breed Chart and select breeds that might suit your goals, then go to Feathersite to look at photos. There are probably a tremendous number of different breeds that will suit you very well. One key to look at is “takes confinement well”.
Henderson’s Breed Chart
http://www.ithaca.edu/staff/jhenderson/chooks/chooks.html
Feathersite
http://www.feathersite.com/Poultry/BRKPoultryPage.html
The “lifestyle” you are talking about is pretty close to what many people in suburbia that can’t free range their chicken have. If they are confined to a run they will pretty much destroy anything green in there. Growing trays of forage sounds like an exercise in futility though. Twelve chickens will destroy any tray you can carry pretty quickly. You’d have to have a whole lot of trays of forage.
I’d suggest a different approach. If you can you might build a frame in the ground maybe 1’ to 2’ wide and a few feet long, then cover that with hardware cloth. The idea is to have it high enough they can peck at the green stuff but not scratch and pull it out. One problem you might have with this approach is that the green stuff would have to establish a good root system so they don’t pull it out of the ground. Grass works really well in something like this.
Or just grow the stuff out of the run and toss it to them. A whole lot of people just dump grass from mowing in there. They love scratching in it and eating it. One potential problem is that you don’t want real long strands of grass. Those can get wrapped up in their crop or gizzard and cause serious problems. Can does not mean it happens all that much but it does happen often enough to be a concern. If you run over the same grass two or three times with your mower it should be in great shape for them. When they peck at grass with roots anchored in the ground, they normally get small bite-sized pieces, not long strands.
Or grow certain plants for them and toss them the leaves or ripe fruit. In your climate I’d think chard or kale would be great. Just cut off some leaves and toss those to them. They will peck those leaves apart and the plants will continue growing new leaves all summer. You might even want to eat some yourself. It doesn’t take much of a garden for chard or kale to produce a lot of leaves. There are lots of other plants that could work for you too.
Or just save your kitchen scraps for them. I don’t know how or what you cook, but they can eat practically anything you can eat. They can also eat almost anything you take off vegetables to prepare them for yourself, things like carrot peels, apple peeling and cores, any greens, anything you cut from tomatoes or peppers, even potato peels. (There is a myth on this forum that they should not eat potato peels. What they should not eat is GREEN potato peels, but then you should not eat the green potatoes either.) So just keep a special bowl in your kitchen and save that stuff for them instead of putting it down the garbage disposal or in the landfill.
I applaud you for wanting to supplement their diet with something green. That adds nice vitamins and minerals for them. But I think those trays will be a lot of work and not real efficient. Hopefully you can get some ideas out of all this that helps.