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Cook pasta as directed on package. Drain well.
Meanwhile, heat 1 tablespoon oil in large skillet on medium-high heat. Add 2 cups of the Swiss chard; cook and stir 1 to 2 minutes or just until tender-crisp. Remove Swiss chard. Set aside. Add remaining 1 tablespoon oil to skillet. Add onion; cook and stir 2 minutes. Add chicken stock and seasonings; bring to boil. Reduce heat to low; simmer 5 minutes.
Place remaining 4 cups Swiss chard in blender container. Add avocado, pistachios and 2 tablespoons of the Parmesan cheese. Add hot stock mixture. Cover blender with lid with center part removed. Cover lid with towel. Blend on low speed 15 seconds. Blend on high speed until mixture is smooth.
Place pasta in serving bowl. Top with Swiss chard sauce, cooked Swiss chard and remaining 2 tablespoons Parmesan cheese. Sprinkle with additional chopped pistachios, if desired.