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.