Using a mandoline or vegetable slicer, cut potatoes into paper-thin slices (about 1/32 in. thick). Coat a 10- by 15-inch nonstick pan with 2 teaspoons oil.
Cover pan bottom with neat, overlapping layers of potatoes, making layers beside pan rim slightly thicker than those in center of pan. With your fingers, gently rub 1 to 2 more teaspoons oil over the top layer of potatoes.
Bake in a 350° oven until potatoes are golden brown (including slices in pan center), 45 minutes to 1 hour; as some areas brown first, cover dark portions lightly with foil. Remove from oven and slide a wide spatula under potatoes to release from pan, but leave in place; (don't worry if the galette cracks). Let cool at least 30 minutes.
Return galette to the 350° oven and bake until slightly darker brown and crisp, 15 to 20 minutes. Remove from oven and slide spatula under galette again to be sure it isn't stuck. Sprinkle potatoes with salt and break into 12 random but similar-size rectangles, saving fragments.