I think the raised bed approach would work well for your backyard. The key is making sure you get enough sunlight throughout the day. Most vegetables need at least six hours of direct sun. You might want to start with tomatoes and peppers since they're fairly forgiving. The soil mix I mentioned before has been working really well for me. I would definitely add some compost a few weeks before planting. Also consider the watering setup because that can make or break everything. Drip irrigation would probably be the most efficient option for your space. Let me know if you want me to help you sketch out the layout. I have some time next weekend if you want to get started on the framing.

The local nursery has a good selection of starter plants right now and their prices are reasonable. I found that starting from seed indoors takes more effort but gives you more variety. For a first project though transplants are probably easier to manage. The soil pH can affect how well things grow so you might want to test that before you start planting.
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The slow braising method really does make a difference with tougher cuts. I usually sear the outside first in a hot pan to get some color then transfer to the oven at around 300 degrees. The liquid should come about halfway up the meat. I like using a mix of broth and a little wine but you can adjust based on what you have. The important thing is keeping the temperature steady and giving it enough time. I usually check after about two hours and then add vegetables if I want them in the dish. The meat should be fork tender when it's ready. I learned that trick from an old cookbook I found at a used store. You can adapt it to just about any protein honestly.

I also think a dry rub with some smoked paprika and garlic powder adds nice depth. Let it rest before slicing to keep the juices where they belong. If you want to thicken the liquid afterward just simmer it on the stove with a little cornstarch slurry. That makes a nice sauce to serve alongside.