This picnic-perfect sandwich is inspired by tortilla Española, a Spanish-style omelet filled with potatoes and onion. We stir in chorizo sausage for savory smokiness and make it portable by serving slices on soft kaiser rolls.
Preheat oven to 375°F. Stir together mayonnaise, Peppadews, and 2 tablespoons scallions; season with salt and pepper. In a medium nonstick skillet, bring 3 tablespoons oil, potatoes, and 1 cup water to a boil over medium heat; season. Boil, stirring occasionally, until potatoes are tender, 8 to 10 minutes. Add remaining scallions and chorizo. Cook, stirring, until scallions are soft, 1 minute. Whisk eggs in a large heatproof bowl; stir in potato mixture to combine. Season with salt and pepper.
Heat remaining 2 tablespoons oil in clean skillet over medium-high. Add egg mixture; cook, stirring to develop curds, 1 minute. Transfer to oven; bake until just set, 10 to 13 minutes. Let cool 10 minutes. Run a rubber spatula around edges; slide onto a cutting board. Slice and serve on rolls, with mayo mixture and arugula.