People ask which recipe to start with. The honest answer is that the pot decides more than the recipe does.
Heavy, with a lid that actually seals
A braise wants an even, gentle heat and its own steam kept inside. Cast iron with an enamelled interior does both: the mass evens out the oven cycling, and the weight of the lid keeps the humidity in. A thin pan with a loose lid gives you scorched patches and a dry surface in the same dish.
Narrow enough that the liquid comes up
The single most common failure is a pot too wide for the amount of food. Liquid should reach roughly two thirds up the meat. In a pot that is too wide it barely covers the base, evaporates in the first hour, and the braise becomes a roast.
If your only pot is wide, use less meat rather than more liquid. Diluting a braise to fill a pot is how you get pale, watery stew.
What you can ignore
Brand, colour, and whether the handles are brass. A £40 pot with a heavy base and a tight lid outperforms a beautiful thin one every time.