Combine the mayo, green onions, lemon juice, garlic & tarragon to make a sauce. Lightly sprinkle the chicken pieces w/salt & pepper.
Unroll the phyllo; cover w/waxed paper & then with a damp kitchen towel to keep it from drying out (I just use a damp paper towel).
Place a sheet of phyllo on a flat surface. Quickly brush w/melted butter (about 2 tsp). Place a 2nd sheet on top of the 1st. Brush w/melted butter. Spread about 1 1/2 TBSP of sauce on each side of a chicken breast (3 TBSP in all.)
Place the breast on one end (corner) of the buttered phyllo sheets. Fold the corner over the breast, then fold the sides over & roll the breast up in the sheets to form a package.
Place in an ungreased baking dish. Repeat with remaining breasts & phyllo sheets, keeping the unused phyllo covered until ready to use it.
Brush the packets w/the rest of the melted butter & spinkle w/Parmesan.
At this point, the dish may be tightly sealed & frozen. Thaw completely before baking. Bake in a preheated 375° oven for 20-25 minutes (more if needed) or until golden.
Serve hot. (serves 12)