Step 1Melt the butter in a small saucepan, stir in the flour, and cook gently over low heat 2 to 3 minutes, stirring frequently. Off heat, pour in the chicken broth and whisk vigorously. Return the sauce to medium-low heat, bring to a simmer, stirring constantly to eliminate any lumps and cook gently, about 5 minutes. Stir in the turkey, green onions, herbs, one-half teaspoon salt and one-eighth teaspoon pepper and let cool, then refrigerate for at least 1 hour.
Step 2Shape the cold meat mixture into four croquettes (roughly 4-inch cylinders flattened at the ends); roll them first in flour, then in egg, and finally in breadcrumbs, making sure that they are coated all over. Chill at least 4 hours or overnight.
Step 3In a large, deep pot, heat the oil until a thermometer inserted reads 360 degrees, or until a crumb of bread dropped in the oil sizzles immediately but doesn't turn dark quickly, and lower the croquettes, one or two at a time, into the hot fat. After about 2 minutes, when browned on the bottom, turn them and fry for 1 or 2 minutes more. Remove them with a slotted spoon and drain on paper towels. Keep warm in the oven until ready to serve, but they are best eaten right away.