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Over low heat, add the olive oil to a medium-sized heavy-bottomed saucepan, then add the potatoes and thyme sprigs and season generously with salt and pepper. Place the lid on the pan and allow to cook, undisturbed, shaking the pan every 5 minutes or so, until the bottoms of the potatoes are browned, about 10 to 12 minutes. (Check occasionally to make sure the potatoes aren't browning too quickly and adjust heat accordingly.) Turn the potatoes to their second side, replace the lid, and continue cooking undisturbed a further 5 to 8 minutes.
Remove the lid and cook a further 2 to 3 minutes, for the condensation to evaporate.
Remove the thyme sprigs and add the sliced garlic and chopped fresh thyme and cook until the garlic is caramelized and chewy, about 2 minutes. Serve hot.