Heat 1 tablespoon butter in 12-inch heavy nonstick skillet over medium-high heat. When foaming subsides, add onion and 1/4 teaspoon salt; stir to coat. Cook, stirring occasionally, until onions begin to soften, about 3 minutes. Reduce heat to medium and cook, covered, stirring occasionally, until onions are light brown and soft, about 12 minutes longer, deglazing with water when pan gets dry, about halfway through cooking time. Transfer to bowl and cover. Do not wash skillet.
While onions cook, microwave 1 tablespoon butter on high power in large microwave-safe bowl until melted, about 45 seconds. Add potatoes to bowl and toss to coat with melted butter. Microwave on high power until potatoes just start to turn tender, about 6 minutes, tossing halfway through cooking time. Toss potatoes again and set aside.
Melt remaining tablespoon butter in now-empty skillet over medium-high heat. When foaming subsides, add potatoes and shake skillet to distribute evenly. Cook, without stirring, until browned on bottom, about 3 minutes. Using spatula, stir potatoes carefully and continue to cook, stirring every 2 to 3 minutes, until potatoes are well browned and tender, 8 to 10 minutes more. Sprinkle with remaining salt and pepper.
Add onions back to skillet and stir to combine; cook until onions are heated through and flavors have melded, 1 to 2 minutes. Transfer to large plate, sprinkle with parsley, and serve.
Yield: 4 servings.