For even cooking and proper browning, the potatoes must be cooked in a single layer and should not be crowded in the pan, so be sure to use a heavy-duty 12-inch skillet. A nonstick skillet simplifies cleanup but is not essential.
If using small potatoes (1 1/2 - 2 inches in diameter) halve each potato. If using medium potatoes (2-3 inch in diameter) quarter each potato to create 3/4 - to 1-inch chunks. Rinse potatoes in cold water and drain well; spread on clean kitchen towel and thoroughly pat dry.
Heat oil in heavy-bottomed 12-inch nonstick skillet over medium-high heat until shimmering. Add potatoes, cut side down in single layer; cook without stirring, until golden brown (oil should sizzle but not smoke), 5-7 minutes. Using tongs, turn potatoes skin side down if using halved potatoes, or second cut side down if using quartered potatoes; cook without stirring until deep golden brown, 5-6 minutes longer. Stir potatoes, then redistribute in single layer. Reduce heat to medium-low, cover and cook until potatoes are tender (Paring knife can be inserted into potatoes with no resistance), 6-9 minutes.
When potatoes are tender, sprinkle with salt and pepper, and lemon chive mixture, and toss or stir gently to combine Serve immediately.