To make the matzo balls, stir together eggs, butter and vegetable stock in a small bowl. Add the matzo meal, parsley, dill and 1/2 teaspoon salt. Stir until evenly combined. Cover and refrigerate for at least 15 minutes and up to 8 hours.
Bring a large pot of water to boil over high heat; add a dash of salt. Using wet hands, for the dough into 12 1-inch balls. Drop the balls, one at a time, into the boiling water. Reduce the heat and simmer for about 30 minutes. Matzo balls should be plump, tender and cooked through.
While the matzo balls are simmering, make the soup. Combine all ingredients except green onions, parsley, salt and pepper in large dutch oven. Bring to a boil over high heat. Reduce the heat and simmer for 20 minutes. Stir in remaining ingredients and cook for another 10 minutes. Remove bay leaf and taste. Adjust seasonings to suit taste.
When matzo balls are done, transfer them to a plate.
To serve, place two matzo balls in a bowl and ladle soup over them.