Preheat the oven to 400 degrees F. Place the potatoes in a pot, cover with cold water and season with salt. Bring to a boil; reduce the heat to medium and simmer until just tender, about 10 minutes. Drain and set aside.
Heat 1 tablespoon olive oil in a medium nonstick ovenproof skillet over medium-high heat. Add the onion; cook, stirring occasionally, until soft, 7 minutes. Add 1 more tablespoon oil, the chard and 1/4 teaspoon salt. Cook, stirring occasionally, until the chard is tender and any liquid has evaporated, 5 minutes; transfer to a large bowl and gently stir in the potatoes, eggs, 1/4 teaspoon salt, and pepper to taste.
Wipe out the skillet. Heat the remaining 1 tablespoon olive oil in the skillet over medium heat. Pour in the egg mixture and top with the soppressata. Cook until the edges start to set, 3 minutes. Transfer to the oven and bake until golden and set, about 12 minutes.
Place the bread on a parchment-lined baking sheet; top with the cheese and parsley. Bake until the cheese is bubbling, 5 minutes. Serve with the frittata.