The competition has traditionally been one reserved for clubs in Leagues One and Two, but this is the third season the country's leading development sides have been allowed to compete.
Leicester's youngsters won a group containing Scunthorpe United, Sunderland and Grimsby Town.
Doncaster finished second in their group, with Rotherham United bottom. They are Yorkshire's only representatives in the knockout stage of the competition.
The League One side won one of their three matches to qualifying, losing the other two.
The last 32 was drawn on a regional basis.
The game is due to take place in the week commencing December 2.
The final is at Wembley in April but in truth, until the arch comes into view, it is seen as something of a nuisance competition for lower-league managers.