Heat broiler. In a large bowl, combine the pork, pine nuts, currants, cinnamon, 1 teaspoon salt, and ¼ teaspoon pepper.
Form the mixture into 20 walnut-size meatballs and place on a foil-lined baking sheet. Broil, turning once, until cooked through, 6 to 8 minutes.
Meanwhile, heat the oil in a large saucepan over medium heat. Cook the garlic, stirring, until golden, 1 to 2 minutes.
Add the chard, ½ teaspoon salt, and ¼ teaspoon pepper and cook, tossing, until wilted. Serve with the meatballs and bread.