Fit potatoes, cut side down, in single layer, in a 12-inch nonstick skillet. Add water, salt, butter, thyme, and garlic to skillet, and bring to a boil over medium high heat.
Reduce heat to medium, cover, and simmer for about 15 minutes, or until potatoes are almost tender.
At this point, remove garlic to a cutting board, and discard the thyme.
Place uncovered skillet over medium to medium high heat, and allow the water/butter mixture to boil slowly until the water evaporates and the butter begins to sizzle and brown. Swirl pan frequently, being sure to keep the cut side of the potatoes in contact with the skillet. (This will probably take 15-20 minutes.)
Meanwhile, after the garlic has cooled a bit, mash the garlic until it forms a paste. (I like to do the same thing that I do with raw garlic. Smash the garlic against a cutting board with the flat side of a chef's knife. Add a sprinkling of kosher salt to help break down the garlic. Once a paste has been created, you're ready to proceed.)
Mix the garlic paste in a small bowl with lemon juice and pepper.
Once the water has evaporated from the skillet and the butter has browned, let the potatoes cook a bit longer, until the browned butter bits stick to the cut surfaces of the potatoes.
Remove from heat, add garlic/lemon juice/pepper mixture and chives, stirring until mixed. Serve immediately.