Combine the shallots and the 3 Tbs. oil in a small saucepan and bring to a simmer over medium heat. Reduce to a low simmer and cook for about 2 minutes, or until the shallots are nicely softened (do not let them brown). Remove from the heat and let cool completely (about 25 minutes at room temperature; cool more quickly in the refrigerator, if you like).
Heat the oven to 400°. Rub the bottom and inside edge of a 9-1/2-inch tart pan with 1/2 tsp. of olive oil or spray with olive-oil spray. Put the tart pan on a rimmed baking sheet lined with foil.
Slice the potatoes as thinly as possible (about 1/16 inch) with a chef’s knife. (Tip: If the potato wobbles, slice a thin sliver off the bottom lengthwise to stabilize it; then continue slicing crosswise.) Discard the ends. Put the potato slices in a mixing bowl, add the shallots and olive oil and the herbs, and toss well to thoroughly coat the potatoes (a small rubber spatula works well).
Cover the bottom of the tart pan with a layer of sweet potato slices, slightly overlapping. Start along the outside edge of the tart pan and, making slightly overlapping rings, move inward until the bottom is covered with one layer of potatoes. Sprinkle some kosher salt (a generous 1/8 tsp.) overall and then sprinkle about a quarter of the Parmigiano and about a quarter of the goat cheese. Arrange another layer of potatoes, season with salt, sprinkle with the cheeses, and repeat two more times, until you have three layers of potatoes. (This is a messy job; you’ll need a damp towel to wipe your hands in between layers.) Top the last layer with more salt and any remaining cheese.
Bake the galette until the top is a deep golden brown (don’t worry if the goat cheese seems very brown) and the potatoes are tender in all places (a fork with thin tines should poke easily through all the layers), 40 to 45 minutes.
Let the galette cool in the pan for 10 to 15 minutes, carefully remove it from the pan, transfer it to a cutting board, and cut it into wedges.