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.