In a 12-inch nonstick skillet over medium, melt 1 tablespoon butter with 1 tablespoon oil. Add the onion and ½ teaspoon salt. Cover and cook, stirring occasionally, until the onion is soft and lightly browned, about 12 minutes, adjusting the heat as needed so the onion cooks evenly. Transfer to a bowl, then stir in the mustard and ¼ teaspoon pepper; set aside. Reserve the skillet.

While the onion cooks, put the potatoes in a colander and rinse under cold running water, using your hands to separate slices. Transfer the potatoes to a medium microwave-safe bowl. Add the remaining 1 tablespoon oil and 3 tablespoons water; toss to combine. Cover and microwave on high until a skewer inserted into a potato slice meets no resistance, 6 to 7 minutes, stirring once about halfway through. Drain the potatoes.

In the same skillet over medium, melt the remaining 2 tablespoons butter. Add the potatoes and ½ teaspoon salt; stir to combine. Cook, turning the potatoes often with a spatula, until lightly browned, about 10 minutes.

Add the onion and stir gently to combine. Cook, stirring once or twice, until the onion is heated through, about 2 minutes.

Off heat, stir in the parsley. Taste and season with salt and pepper and additional mustard, if desired.