In a medium bowl, combine the mayo, green onion, lemon juice, garlic, and tarragon. Blend well.
Lightly sprinkle chicken with salt and pepper.
Place a sheet of phyllo on a working surface. Quickly brush it with melted butter. Spread about 1½ tablespoons of the mayo mixture on each side of a chicken breast. Place the breast in one corner of the buttered phyllo sheet. Place another buttered phyllo sheet on top, and fold a corner over the breast. Fold the sides over and roll up the breast in the sheets to form a package. Place it in an ungreased baking dish. Repeat with the remaining breasts and phyllo sheets.
Brush the packets with the rest of the butter and sprinkle with Parmesan.
Bake 30 to 35 minutes, until golden.
Notes
*Pound the chicken breasts with meat tenderizer to even out thickness.
*Phyllo is sometimes tricky to work with- be sure to cover with a damp towel so they do not dry out.