In a small bowl, combine the onions, mayonnaise, lemon juice, 2 garlic cloves and tarragon; set aside. in another small bowl, combine the butter and remaining garlic.
Place one sheet of phyllo dough on a work surface with a short edge facing you. Brush with 2 teaspoons butter mixture; brush to distribute evenly. Repeat with one more sheet of phyllo, brushing with another 2 teaspoons of butter mixture (Keep remaining phyllo dough covered with plastic wrap to avoid drying out).
Lightly sprinkle chicken breasts with salt and pepper. Center one chicken breast on the lower third of phyllo. Spread about 3 tablespoons of mayonnaise mixture over chicken breast. Fold bottom edge over chicken, then fold in sides. Roll up jelly-roll style; cover with plastic wrap and set aside. Make five more chicken packets.
Place packets in an ungreased 15x10x1-in. baking pan. Brush tops with remaining garlic butter; sprinkle with cheese. Bake uncovered, at 375° for 25-30 minutes or until a thermometer reads 170°. Serve warm.
Editor’s Note: Reduced-fat or fat-free mayonnaise is not recommended for this recipe.