Remove meat from chicken and discard skin and bones. Chop chicken into large diced pieces. Set aside.
Heat butter in a large pot over medium-low heat. Add the onion and cook for 10 to 15 minutes, until translucent. Add 1 to 2 tablespoons water if the pan becomes too dry.
Add the cornstarch and cook over low heat, stirring constantly, for 2 minutes. Be sure there are no lumps.
Add chicken stock, sherry and heavy cream. Bring to a boil then reduce to a simmer.
Add the diced chicken, carrots, peas, onions, and tarragon. Simmer for 5 minutes.
Ladle into 4 ovenproof bowls.
unroll the phyllo dough and cut it into quarters. Place a quarter-sheet of phyllo on top of each ovenproof bowl and brush with olive oil. Repeat layering with all three layers. The edges of the phyllo will extend over the sides of the dish; trim if desired.
Place bowls on a rimmed baking sheet and bake 25 to 30 minutes.