The 19-year-old, who had been linked with a move away from south Yorkshire this summer, was named as the club's Young Player of the Year this term.
"Jamie has been outstanding this season and we are very pleased that he has committed himself to the club for the next two years," said chief operating officer Paul Douglas.
"He has more than earned his extended deal and we hope he will remain with the Club for many years to come.
"We are very proud of our youth team and Jamie, along with Stephen Brogan and Ryan Taylor, are all local graduates making big strides in our first team.
"To blood so many home grown players over recent years is a fantastic achievement, and credit must go to those involved in the youth system as they continue to provide the first team with some exceptional talent."