Braise the cut in a cast iron dutch oven.

https://www.allrecipes.com/article/how-to-braise-meat/

Slow and low. As someone mentioned below, keeping the meat covered while cooking in a dutch oven or tall roasting dish is crucial for making sure that it doesn’t dry out. In a pinch the crappy aluminum roasting pans will work just fine, the key is that your roasting container should be taller than the meat do that you can cover it while it’s cooking. For beef and pork, around 200 Fahrenheit and for chicken thighs, around 275. Chicken breasts don’t have enough fat as it is white meat so I wouldn’t try them slow and low. Just cook until it falls apart to your liking, time depends on the size and cut but those temperatures will do you just fine.

Marinade the meat for at least 12h in a mix of some random fat, random acid, salt, and seasoning. Then roast it in an oven bag. Slow and low heating (I use 125°C), trying to minimise the amount of vapour that is formed. It’ll take a few hours, and you need to flip it halfway so both sides get in contact with the juices. Near the end, if you want, open the bag and rev up the oven temperature, to brown it on the outside and, if there’s leftover juice, let it dry and coat the meat.

Which acid is up to you; I use alcohol vinegar if I don’t want it to interfere too much on the taste, otherwise I use a red dry wine for beef and lime juice for pork. The “random fat” is usually veg oil, just so the seasoning spreads more evenly across the meat.

Although… frankly, I’d rather use a pressure cooker for those.

Low and slow is my advice too - but also IME there’s something magical that happens when you use creamy soups as a starter with your meats. I spent years trying to make something healthier than my mom’s roast beef recipe, which is basically cream of mushroom soup and onion soup mix plus a roast, but nothing I did could make it fall apart and stay moist like hers. I settled on using real onions in place of the onion soup mix which helps lower the salt content a bit but my family often asks me to just make it like my mom did.

For Pork Shoulder, I found Michael Rhulmans recipe to be pretty versitile:

To cook the pork, as noted in post: Ideally, sear it hard over coals in the Weber, covered for 20 to 30 minutes, then put it in a Dutch oven, covered, for 6 hours at 250 degrees (or 4 hours at 300˚F). I think smoke is critical, but if you want to make it super easy on yourself, put the raw shoulder in a Dutch oven and roast it covered at 225 overnight and that’ll do the trick as well. Stir in the sauce. Taste for seasoning—salt, sweetness, acidity,heat. Adjust as you wish

My technique is to “shock” the meat (not sure of the right term here) and then turn the heat way down and go low and slow… So for example… If I have a 5 lb shoulder roast I’ll go 1/2 hour uncovered at 475F and then turn it down to 200-250 and go another 4 or so hours (still uncovered) or til it’s about 165 internal on a meat thermometer. The 1/2 hour at 475 forms the crust and seals in the juices.

Definitely tent the meat for a few hours while cooking.

I do a Sous vide at 160 for 18 hours or so. Meat falls apart and is incredibly easy to make. Just finish under the broiler and you’re good to go.

Probably pretty different than what you’re looking for, but it definitely falls apart for pork shoulders.

https://www.seriouseats.com/sous-vide-carnitas-crispy-mexican-style-pulled-pork-recipe

The simple answer is braising. Low and slow, it will work.

long cooking, my kids pork neck meat made like this:

• layer of chopped leek (thin half slices?)
• layer of pork neck meat (chunks/cutlet size slices)
• layer of button mushrooms/champignon/portobello whatever you call it in your part of the world, at least halved, i like to slice them to make a few equal pieces each
• pour thin 30% cream on it (it’s called whipping cream in US?), it’s important for it to be thin and go between the rest of the ingredients, if you add thick one it’ll stay on the top, burn and smell weird
• of course season accordingly, put it into the oven for 3 hours at 180℃ covered, uncover for last 20-30minutes

no need to sear the meat before, tried it and there was no difference, only wasted time, one time i was time pressured to take it out an hour early and it was edible of course but hard

Fat and connecting tissue melts with heat, that’s what gelatin is.

Time and heat=fallapart dry meat. Balance heat and time to have it fall apart and not be dry, but you can’t perfectly have both. Sous vide, lowest possible foodsafe temp and long time works pretty good, but you still don’t want to go overboard.

Also, everybody likes the mallard reaction and browning and you don’t get that with low heat (like 129F).

If all you want is fall-apart meat, that’s easy, though, just render all the connective tissues into liquid.

Dutch oven in an oven, lid on, 250°F for 3-5 hours. Perfect for a 3 lb corned beef brisket, probably works on other things.

Marinade, then slow cook in pan with foil on low heat. I can almost taste the brisket just writing this.