Leave the potatoes whole if less than 1 inch in diameter; halve them lengthwise if fatter. Arrange the potatoes in a single layer in a 12-inch skillet. They will be crowded, but they shouldn’t be stacked. Tuck the thyme sprigs between the potatoes. Cut 3 Tbs. of the butter into 3 pieces. Add the butter, 3/4 tsp. salt, and a few generous grinds of pepper to the potatoes.
Pour over just enough water to almost cover the potatoes (about 2 cups). Partially cover and bring to a gentle simmer over medium heat. Simmer, stirring gently once or twice, until the potatoes are tender when pierced with a skewer, about 25 minutes. (If the water threatens to dry up before the potatoes are tender, add another 1/2 cup.)
Transfer the potatoes to a serving dish with a slotted spoon. Increase the heat to high and boil the remaining liquid, uncovered, until it’s reduced to a buttery glaze, 5 to 8 minutes, depending on the amount of liquid. Remove the thyme sprigs (most of the leaves will have fallen off). If you would like a richer sauce, swirl in the remaining 1 Tbs. butter. Pour the glaze over the potatoes and serve immediately.