Preheat the oven to 300°F . Prick each yam a few times with a fork. Set in a baking pan and bake until just tender, about 50–60 minutes depending on size. Meanwhile, melt the 2 Tbsp. of the butter in a small pot over medium-low heat. Add the onion and garlic and cook, stirring, about 10 minutes, until golden and sticky. Remove from heat and set aside.
When cooked, cool the yams until safe enough to handle. Now carefully remove the skin with a small knife. Slice each yam into 9 even rounds. Season each round with salt and pepper. Spread or crumble goat cheese on one side of 12 of the yam rounds. Place 6 of the goat cheese topped yams in a parchment-lined baking dish. Top them with half the onion/garlic mixture.
Set the six other goat cheese topped yams on the onion-topped ones, gently pressing each one down. Top those yams with the remaining onions and then set on the 6 remaining yam slices. Drizzle the top of each yam with the 1 Tbsp. melted butter. (Yam stacks can be made to this point many hours in advance. Cover and refrigerate until ready to bake.) Bake yam stacks in a preheated 300°F oven for 15 minutes, or until yams, cheese and onions are heated through.
Note: Yams that are about 9” long and 3” wide are best for this recipe.