Cut out the chard stems, dice, and set aside. Coarsely chop the chard leaves.
In a large cast iron skillet (or a skillet with an oven proof handle), heat 1 tablespoon olive oil. Add onion, ¼ teaspoon of salt, and rosemary. Sauté for 10 minutes over medium heat, until the onions are soft and beginning to brown.
Stir in the chard stems and garlic, sauté for a few minutes, then add the chard leaves and ¼ teaspoon salt, and sauté for another 1 to 2 minutes, until the greens begin to soften. Remove from heat and set aside.
In a large bowl, whisk together the eggs. Add the chard mixture and goat cheese, and stir to combine.
Preheat the broiler in the oven. Heat 1 tablespoon of oil in the skillet over medium-high heat.
Once the oil is hot, pour in the chard-egg mixture and cook for about 3 minutes. Place the skillet in the oven under the broiler for 3 minutes.
Fry the frittata from the edges of the skillet and flip it onto a serving platter. Cut into wedges and serve. Add fresh ground pepper to taste.