Preheat oven to 375 degrees F. In a small bowl, combine butter with sugar, cinnamon and salt, combining well until throughly mixed. Place butter on a sheet of plastic wrap and form it into a small rectangle or square. Place in the freezer for 20-30 minutes, until firm.
Very carefully (and um, patiently) slice the sweet potatoes every 1/8 to 1/4 inch and only 3/4 of the way down the potato. I found it helpful to slice the potatoes halfway down the whole way across, then once that was done carefully use a smaller pairing knife and slice deeper until you almost reach the bottom. Remove the butter from the freezer and slice very, very thin pieces, then cut those in half. Ideally you want about about 24-30 small, thin pats of butter. Gently pull apart the sweet potatoes and place a butter pat in every three or four slices. Sprinkle potatoes with additional cinnamon if desired and place in a baking dish so they sit upright.
To make the crumble, combine the sugar, oats, cinnamon and salt and mix well. Using your hands, crumble in the butter and mix until the entire thing is moistened and comes together. Place an equal amount of crumble on top of each potato, pressing firmly but gently. Bake for 30-35 minutes, then cover the potatoes with foil and bake for another 20-25 minutes. To serve, cut in half.