Cover potatoes generously with well-salted cold water in a large pot and bring to a boil. Reduce heat and simmer potatoes, partially covered, until they are just barely tender, 20 to 25 minutes.
Transfer potatoes to a colander using a slotted spoon, rinse under cold water to stop the cooking, and dry.
Chill potatoes on a plate, uncovered, at least 8 hours.
Peel potatoes and coarsely grate into a large bowl.
Add thyme, 1 1/2 teaspoons salt, and 3/4 teaspoon pepper, then drizzle with 2 tablespoons butter and toss mixture well.
Heat oven to 375°F with rack in middle.
Heat oil and 1 tablespoon butter in skillet over medium-high heat until foam subsides, then add potato mixture and tamp it down firmly in skillet with a rubber spatula, smoothing the top.
Reduce heat to medium and cook potato cake, covered with a lid, until edges begin to turn golden, about 5 minutes.
Remove lid, then bake potato cake, uncovered, 10 minutes.
Brush top of potato cake with remaining 2 tablespoons butter.
Turn on broiler and broil potato cake, 4 to 5 inches from heat, until golden brown, 3 to 7 minutes.
Remove skillet from oven carefully (wear oven mitts!). Run the rubber spatula around the edge of the cake to loosen it from the skillet.
Invert a large round heat-proof plate over skillet and invert potato cake onto plate.
Cut into wedges and serve.
•Potato cake can be made 1 day ahead. Cool, then wrap—still on the plate—in wax paper followed by foil, and chill. To reheat potato cake, remove foil and paper and bake in a 350°F oven until heated through, about 20 minutes.
