Brilliant Brook's 132 puts England on top against sloppy New Zealand
At stumps Brook was 132 not out with the visitors recovering from 71-4 to be 319-5, trailing the hosts by 29 runs and with five wickets in hand.
Captain Ben Stokes was unbeaten on 37 after being dropped by Tom Latham on 30 -- New Zealand's sixth dropped catch and the third by their skipper.
After England's woeful start the 25-year-old Brook, in partnership with Ollie Pope, went on the attack.
Aided by luck from poor fielding and several edges sailing just wide or over slips, the pair belted 151 off 171 deliveries for the fifth wicket.
"From where we were to where we are now, the boys are very happy," Pope said.
New Zealand debutant Nathan Smith said it was vital the home side tidied up their fielding when they take the new ball on Saturday.
"That wicket has enough bounce in it and with the new ball as well it will nip a little sharper," he said.
"It's really key that we're nice and accurate with the new ball and take the chances when they come."
Brook was dropped four times -- on 18, 41, 70 and 106 -- with the last detected by television replays after a spill by wicketkeeper Tom Blundell.
England's batting onslaught came as the cloud cover which aided the New Zealand seamers in the first session gave way to blue skies.
New Zealand, 319-8 overnight, were all out for 348 and then reduced England to 45-3 by lunch.
When Ben Duckett went for 46 after lunch England were on the ropes at 71-4, but it would be 30 overs before New Zealand claimed their next wicket.
Pope was on 77 with eight fours to his credit when he slashed at a wide delivery from Tim Southee and a diving Glenn Phillips grabbed a stunning one-handed catch.
Phillips threw his arms in the air in celebration, in stark contrast to when he put his head in his hands earlier after Devon Conway dropped Brook off his bowling.
England grab momentum
In the early cloudy conditions the New Zealand seamers were in their element with plenty of movement.
A Matt Henry delivery that nipped in trapped Zak Crawley lbw without scoring.
Smith followed with an angled delivery to Jacob Bethell which was edged to Blundell and England's debutant was gone for 10.
Four balls later, of which two were no-balls, a similar angled delivery cramped Joe Root -- in his 150th Test -- for room and it deflected off his pads and on to the stumps for a duck.
Duckett refused to be subdued and chanced his arm at anything loose.
He was dropped by Latham on 23 and survived two inside-edges before he pulled a rising Will O'Rourke delivery to Conway waiting on the boundary.
In the following over Brook was dropped for the first time and momentum began to shift in England's favour.
The partnership with Pope galloped along at five an over and Brook brought up is seventh Test century steering a Tim Southee delivery past backward point to the boundary.
New Zealand had added a further 29 runs to their overnight score, the bulk of them by Phillips.
He ended unbeaten on 58, his fifth half-century and the second-highest New Zealand score behind Kane Williamson's 93.
Brydon Carse finished with 4-64, the best figures for England, after taking both wickets to fall on the second morning.
With his first ball of the day he had Southee caught on the boundary for 15, then ended the hosts' innings by bowling O'Rourke for a duck.