Look for baby Yukon Gold potatoes that are about 1 1/2 inches in diameter. If you can't find them, try small boiling potatoes.
Preheat oven to 400 degrees. Place garlic on parchment-lined foil. Season with salt and pepper, generously drizzle with oil, and wrap. Roast until fragrant and soft, about 1 hour. Leave oven on.
Cover potatoes with salted water by 2 inches in a medium saucepan. Bring to a boil, and cook until fork tender, about 20 minutes. Drain, and return to hot pan. Let stand for 5 minutes to dry.
Halve potatoes crosswise. Trim (and reserve) rounded ends of each half slightly, so potatoes sit flat with cut sides up. Scoop out flesh using a melon baller, leaving a thin layer on bottom and sides, and transfer to a bowl. Peel reserved ends; add to bowl. Stir in oil, sour cream, and thyme. Squeeze garlic cloves from skins into potato mixture. Mash until smooth. Season with salt and pepper.
Lightly brush a baking sheet with oil. Fill each potato half with about 1 1/4 teaspoons potato mixture, mounding slightly. Place potatoes 1 inch apart on sheet, and lightly drizzle with oil. Bake until golden brown, 25 to 30 minutes.
Bring to room temperature, and bake just before serving.
Potatoes can be stuffed and refrigerated overnight. Bring to room temperature, and bake just before serving.