Ian Westwood hit a career best 196 as Warwicks made 435 all out on the second day of their County Championship game with Yorkshire at Headingly.
Resuming on 270-4 with Westwood on 151, McKay went for seventeen before Westwood was out with the score on 360 when he was trapped lbw by Middlebrook after facing 447 and four short of a double century.
Ricky Clarke added 39 and Keith Barker added twenty seven before falling to Middlebrook.
Jeetan Patal added 31 and when Boyd Rankin was out for three and Middlebrook's third victim, Warwicks were all out for 435.
Yorkshire batted and in the sixteenth over, Patal had Lees out lbw for 19 with the score on 41. It was 88 when Patal had Will Rhodes, who had made 46 caught by Rankin.
Before stumps, Rankin had Pujara caught behind by McKay as Yorks closed on 128-3.