Arrange potatoes in single layer, cut side down, in 12-inch nonstick skillet. Add water, butter, garlic, thyme, and salt and bring to simmer over medium-high heat. Reduce heat to medium, cover, and cook until potatoes are just tender, about 15 minutes.

Remove lid and use slotted spoon to transfer garlic to a cutting board; discard thyme. Increase heat to medium-high and vigorously simmer, swirling pan occasionally, until water evaporates and butter starts to sizzle, 15 to 20 minutes. When cool enough to handle, mince garlic to paste. Transfer paste to bowl and stir in lemon juice and pepper.

Continue to cook potatoes, swirling pan frequently, until butter browns and cut sides of potatoes turn spotty brown, 4 to 6 minutes longer. Turn off heat, add garlic mixture and chives and toss to coat.