Arrange the potatoes in a single layer, cut side down, in a 12-inch nonstick skillet. Add the water, butter, garlic, thyme, and salt and bring to a simmer over medium-high heat. Once simmering, reduce the heat to medium, cover, and cook until the potatoes are just tender when pierced with a fork, about 15 minutes.

Remove the lid and use a slotted spoon to transfer the garlic to a cutting board; discard the thyme sprigs. Increase the heat to medium-high and simmer vigorously, swirling the pan occasionally, until the water has completely evaporated and the butter starts to sizzle, 15 to 20 minutes. While the water evaporates, mince the softened garlic into a smooth paste and stir it into a small bowl with the lemon juice and pepper.

Continue to cook the potatoes in the remaining butter, swirling the pan frequently, until the butter solids turn brown and the cut sides of the potatoes are spotty brown and crisp, 4 to 6 minutes. Remove from the heat, add the chives and the garlic-lemon mixture, and toss to coat every potato thoroughly. Serve hot while the skins are still crisp.