Preheat the oven to 400°F. Strip the kale leaves from the stems and coarsely chop the leaves. Thinly slice the stems and keep them separate from the leaves.
Pour 1 tablespoon of the oil into a medium skillet over medium heat. When it starts to shimmer, sprinkle in the pimenton and let it sizzle and bloom for a few seconds, then add the onion, garlic, and sliced kale stems and sauté until tender. Add the mushrooms and sauté until they collapse and release their liquid, then add the kale leaves and continue cooking until the liquid has evaporated. Season with salt to taste and remove from the heat.
Pour the remaining tablespoon of oil into a small, well-seasoned cast-iron skillet over medium heat. Carefully arrange half of the sweet potato slices in the skillet in concentric circles, overlapping to form a couple of layers; sprinkle each layer with a little salt as you go. Spoon on the mushroom-kale mixture, and top with the grated cheese.
Arrange the remaining sweet potato slices on top, sprinkling each layer lightly with salt as you go. Press the galette with a spatula, cover the skillet tightly with aluminum foil, and bake until the sweet potatoes are easily pierced with a fork, 20 to 25 minutes.
While the galette is baking, sprinkle the pecans into a small skillet over medium-high heat. Cook, shaking the pan frequently, until the nuts start to brown and become fragrant, a few minutes. Immediately transfer them to a plate to cool; if you leave them to cool in the pan, they can burn. Once they are cool, chop them.
Remove the galette from the oven and take off the foil. Turn the oven to broil and slide the skillet under the broiler element or flame until the sweet potatoes just brown on top.
Let the galette cool for a few minutes, then run a knife around the edges of the skillet to loosen it. Invert a plate over the skillet and, using oven mitts, hold the skillet and plate together and quickly flip the two so the plate is on the bottom and set it on the counter. Lift off the skillet. Some of the potato slices may stick to the pan; use a spatula to scrape them out and patch up the galette.
Sprinkle with the green onion slices and nuts and eat. (If you prefer, you can leave the galette in the pan and cut wedges out of it for eating.)