Thinly slice the Swiss chard leaves and cut the stems into 1/4-inch slices. Heat the olive oil in a large skillet over medium-high heat. Add the onion and chard stems, season with kosher salt and pepper and cook, stirring occasionally, until the onion is soft and just beginning to brown, about 10 minutes. Add the dill and lemon zest and cook, stirring, 30 seconds. Add the chard leaves and cook, stirring, until wilted, about 4 minutes. Remove from the heat and stir in the parsley, 3/4 teaspoon kosher salt, and pepper to taste. Let the filling cool, then squeeze out the excess moisture.