Regarding the question about braising tougher cuts, you will want to sear the meat first on high heat until you get a deep brown crust on each side. Transfer to a heavy pot with your aromatics and enough liquid to come about a third of the way up the meat. Keep the temperature low and steady, around 300 degrees, and give it several hours. The collagen breaks down slowly and that is what gives you the silky texture. You can check by piercing with a fork - if it slides off easily, it is done. Add your vegetables later in the process so they do not become too soft or mushy. Let the meat rest before serving and consider skimming the fat from the cooking liquid if you want a cleaner sauce. I usually let it cool slightly and then spoon off the fat. The liquid reduces beautifully on its own but you can thicken it with a bit of flour if needed. Let me know how it turns out when you try it.

On the composting question, you are right that balancing green and brown materials makes a noticeable difference. I aim for roughly equal parts by volume, though it does not have to be exact. The key is turning the pile regularly to keep air circulating throughout the mixture. If it starts to develop an odor, you probably need more browns like dried leaves or shredded paper. Keep the pile as moist as a wrung-out sponge and in warmer months the process speeds up considerably. I have found that chopping materials into smaller pieces helps them break down faster and more evenly. You can start a simple pile in a corner of the yard without any special bin if you want to try it out first. A pitchfork works well for turning and you will be surprised how quickly it shrinks down once everything gets going. Let me know if you want suggestions on what to layer first.