Notes: You can complete steps 1 through 3 up to 1 day ahead. Chill sweet-potato mixture airtight; cover shallots and store at room temperature. Reheat sweet-potato mixture in a microwave oven, stirring occasionally, for 3 to 4 minutes.
1 1/2 pounds sweet potatoes (1 1/2 to 2 in. wide)
12 ounces shallots (3/4 to 1 in. wide), peeled
2 tablespoons olive oil
1 1/2 tablespoons balsamic vinegar
About 1/2 cup milk
3/4 cup fresh chèvre (goat cheese)
2 tablespoons butter
1 teaspoon minced fresh thyme leaves or dried thyme
Scrub sweet potatoes, pierce each in several places with a fork, and set in an 8- or 9-inch-wide pan. In another pan, mix shallots with oil and vinegar.
Bake shallots in a 375º oven, shaking pan occasionally, until deep golden brown, about 45 minutes; bake sweet potatoes until they give very easily when squeezed, about 1 hour.
Protecting your hands with a potholder, slit hot sweet potatoes open and scoop flesh into a bowl; discard skins. With a potato masher or a mixer, mash sweet potatoes, 1/2 cup milk, cheese, butter, and thyme until smooth; for a softer texture, beat in more milk. Season to taste with salt and pepper.
Lay a large, hot galette piece on each of four warmed dinner plates. Mound half the hot sweet-potato mixture equally in center of galettes.
Lay another galette flat on each sweet-potato mound. Top equally with remaining sweet-potato mixture. Stick remaining galette pieces and any fragments vertically into the sweet-potato mixture; spoon shallots around napoleons.