England started batting to reach their 277-run target set by New Zealand to win the first Test at Lord's on Saturday.
World Test champions New Zealand, who had been 251 for four, lost their last six wickets for 34 runs on the third morning -- three falling to successive Broad deliveries.
England, however, face a tough task if they are to go 1-0 up in this three-match series and make a winning start under their new leadership duo of captain Ben Stokes and coach Brendon McCullum, the former New Zealand skipper.
There have been only three successful fourth-innings run chases of more than 200 runs in Lord's Test history, the most recent when England made 282 for three against New Zealand in 2004.
After rain delayed Saturday's scheduled start by 30 minutes, New Zealand resumed on 236 for four in their second innings, a lead of 227.
Mitchell was 97 not out overnight and Tom Blundell unbeaten on 90 after they had come together Friday with the Black Caps in trouble at 56 for four.
Mitchell wasted no time in reaching his second century in 10 career Tests, off-driving Broad for three from his first ball of Saturday's play.
The 31-year-old only playing in this match after Henry Nicholls was ruled out through injury, went to three figures in 189 balls including 11 fours.
After waiting 366 deliveries to take the fifth wicket, England astonishingly collected three in three balls for a team hat-trick.
Broad, recalled along with James Anderson after England's two all-time leading Test wicket-takers were controversially left out of a series loss in the Caribbean, was the man to make things happen.
He dismissed Mitchell and Kyle Jamieson, either side of Ollie Pope running out Colin de Grandhomme.
Mitchell was caught behind for 108 off a Broad delivery that nipped away to end a New Zealand record fifth-wicket partnership against England of 195, surpassing the 180 shared by Martin Crowe and Shane Thomson at Lord's in 1994.
Broad's next ball led to a huge lbw appeal against de Grandhomme, who carelessly wondered out of his crease and was run out by Pope's direct hit from the slip cordon before Broad bowled Jamieson for a golden duck.
Anderson then got in on the act by removing Blundell, who had been watching the wickets go down from the other end while taking his score to 96.
But he fell short of a place on the honours board when Anderson had him plumb lbw before Matthew Potts, who took four wickets for 13 runs in New Zealand's first innings 132, continued his fine debut by removing Ajaz Patel.
Concussion substitute and England debutant Matt Parkinson, summoned after Jack Leach was injured fielding Thursday, ended the innings with his first Test wicket when the leg-spinner had Tim Southee caught at slip.
BDST: 1855 HRS, June 4, 2022
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