Bradshaw continued a fine run of form by giving the Reds the lead in the opening 10 minutes, before adding a sublime second after Everton had levelled.
His brace looked likely to be enough to secure the three points for Marc Bridge-Wilkinson’s side, but two goals in the last few minutes turned the game in favour of the home team.
It was an entertaining U18 Premier League North contest in the winter sunshine, especially during a first half in which there were chances at both ends from the start.
Everton’s Malik Olayiwola was the first to test out a goalkeeper with a low drive that Bailey Hall was behind easily, while a good run from Oliver O’Connor for the Reds could have produced an assist for Bradshaw but he never quite found the pass.
It did not take long for Bradshaw to have his first big impact on the match, however, as he gave the visitors the advantage in the eighth minute.
Clae Ewing’s ball into the box was blocked but played up into the air by the overlapping Harry Evers, and Bradshaw was there at the back post to head home.
It was a lead that should have lasted just a minute; it took an astonishing point-blank save from Hall to deny a header from George Morgan and maintain the advantage.
In an exciting passage of play, Liverpool went straight up the other end and Josh Sonni-Lambie seemed to have made it two with a fine finish, only for the flag to be raised for a narrow offside.
Everton’s Braiden Graham was booked for a dive as he tried to go around Hall, though he made up for it shortly afterwards with the equaliser for Everton. The Reds lost the ball as they played out and Graham was on hand to place a shot past Hall for 1-1 on 22 minutes.
Another lovely run from O’Connor released Bradshaw on the right and his pass set up Ewing for a shot that was blocked. However, just after the half-hour they did restore their lead courtesy of an excellent solo effort from Bradshaw.
He ran at defenders into the box, produced a drag-back to the byline and then finished across goal into the far corner from a narrow angle – much to the delight of the watching fans.
Joel Catesby blazed over for Everton and Liverpool captain Lucas Pitt cleared a Morgan effort off the line and into the grateful arms of Hall before the break.