Combine the shallots and 3 Tbs. of the oil in a small saucepan and bring to a simmer over medium heat. Reduce to a low simmer; cook the shallots until nicely softened (don’t let them brown), about 2 min. Remove from the heat and let cool completely (about 25 min. at room temperature; cool them more quickly in the refrigerator. if you like).
Heat the oven to 400°F. Rub the bottom and inside edge of a 7-1/2-inch tart pan with a removable bottom with the remaining 1/2 tsp. 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 lengthwise sliver off the bottom to stabilize it; then continue slicing crosswise. Discard the ends. Put the potato slices in a mixing bowl, add the shallots and olive oil along with 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 potato slices, overlapping them slightly. 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 the potatoes with salt (a generous 1/8 tsp.) and then sprinkle about one-quarter of the Parmigiano and about one-quarter of the Gruyère over all. Arrange another layer of potatoes, season with salt, sprinkle with cheese, and repeat two more times, until you have four layers of potatoes. (This is a messy job; you’ll need a damp towel to wipe your hands between layers.) Top the last layer with more salt and any remaining cheese.
Bake the galette until the top is a reddish golden brown and the potatoes are tender in all places (a fork with thin tines should poke easily through all the layers), 45 to 50 min. The bottom will be crisp and the sides brown.
Let the galette cool for 10 or 15 min. in the pan. It will then be cool enough to handle but still plenty hot inside for serving. Have a cutting board nearby. Run a paring knife around the edge of the galette to loosen it and carefully remove the tart ring by gently pressing the tart bottom up. Slide a very thin spatula under and all around the bottom layer to free the galette from the tart bottom. Use the spatula to gently slide the galette onto a cutting board. Cut into four or six wedges, or as many as you like.
Individual Potato Galettes
For a special dinner, you can make potato galettes in individual tart pans. The amount of ingredients in the recipe will fill (with a little to spare) four 4-1/2-inch tart pans with removable bottoms. Baking times will be just slightly shorter, but remember, you can bake these ahead, remove them from the tart rings, and reheat them on a baking sheet. A mini galette makes a lovely bed for a slice of beef tenderloin with a little sauce.