Add the bread slices to a food processor and pulse for 30 seconds to make breadcrumbs.
Transfer the breadcrumbs to a bowl. Add the onion, ground chicken, egg, black pepper, salt, nutmeg, and allspice powder. Mix until well combined.
Use an ice cream scoop to portion the meatball mixture and roll each portion into a ball. Repeat with the remaining mixture.
Heat the oil in a heavy-bottomed skillet over medium heat. Add the meatballs and cook for 5 to 6 minutes, turning them frequently for even cooking.
Transfer the meatballs to a plate and discard any excess moisture or oil from the skillet.
Add the butter to the same skillet. Once the butter has melted, sprinkle the flour over it and cook for 1 minute to form a roux.
Slowly add the chicken broth and heavy cream to the roux, whisking until combined. Simmer the gravy for 4 to 5 minutes.
Return the chicken meatballs to the skillet and cook for 4 to 5 minutes, until the gravy thickens and the meatballs are cooked through.Note: Use a meat thermometer to check for doneness. The chicken meatballs should reach an internal temperature of 165°F at the center before serving. Garnish the Swedish chicken meatballs with chopped parsley and serve over mashed potatoes with vegetables on the side.