Heat oil in a large saucepan over medium heat. Add onion and garlic and cook, stirring often, until translucent and soft (do not brown), 7-8 minutes. Stir in chard, broth, parsley, cilantro, fresh and dried mint, and nutmeg.
Bring to a boil, reduce heat, and simmer, stirring occasionally, until chard is tender, about 10 minutes. Stir in lemon juice and season to taste with salt and pepper. Working in batches, purée soup in a blender until smooth. Return to pan. DO AHEAD: Can be made 8 hours ahead. Cover and chill. Rewarm soup before continuing.
Place 1/3 of yogurt in a medium bowl. Add 1/2 cup warm soup; whisk until smooth. Repeat process twice more, adding a total of 1 cup more soup. Whisk yogurt mixture into soup in saucepan. Stir 1/4 cup herbs and half of feta into soup. Season to taste with salt, pepper, and lemon juice, if desired.
Ladle soup into bowls and garnish with remaining 1/4 cup herbs and 2 oz. feta. Drizzle with oil, if desired