We make reestit mutton often and I do love it (never tried it before I moved up here) - we use cuts of leg joints from two year old or older shetland sheep. Prepare a large tupper ware box full of salted water - this next bit is going to sound really weird but there is scientific purpose to it. To get the correct salt ratio to water you have to fill the container with warm water and start adding common table salt, keep stirring until dissolved then float a large egg size raw potato with a 3" nail in it (this gives the correct weight) if it sinks then keep adding a cupful of salt, dissolve and try floating again, I also add a couple of cups of sugar to balance the flavour. This needs to be done the day before and then chilled so the water is completely cold before the meat goes in. Once your tattie is floating at the top of the water add your cuts of leg - don't bother with the shank, these need to be weighted down into the liquid, we use a glass casserole lid then cover with lid and put in the fridge for 3 weeks. Then remove the cuts rinse, dry with kitchen roll and hang to dry for a couple of days - we hang on our pulley above the rayburn. After that you can freeze or cook by bringing to the boil and then simmering very gently for a couple of hours - I'll add black pepper bay leaves and or cardamoms, cumin etc for extra flavour. Then you can eat hot with mash and neeps, and try cold the next day sliced very thinly - fantastic with homemade oatcakes or one of my favourite ways as a soup with leeks and potatoes.
There is a butcher in Shetland who makes lots of the stuff and I've eaten his stuff raw, when sliced thinly enough tastes just like prosciutto but lamby - lovely!
The traditional way was to wait until the cold winter weather came and folk would do a large amount of cuts into barrels full of salt water, but we just don't get that cold up here these days so a fridge is the only option.