The 17-year-old Lawrence scored 161 from 206 balls as Essex compiled a mammoth 610-8 before declaring, Nick Browne adding 143. Graham Napier added some tail-end wag with 73.
Surrey made 340 from their first innings and were 175 behind at the close of play after totting up 95 for two.
Kent were made to follow on against Lancashire at Old Trafford and finished the third day 44 runs behind on 148-3.
Opener Joe Denly led the resistance before being dismissed for 60 and captain Rob Key was bowled by Peter Siddle (1-17) for 34.
Derbyshire remain in control of their match against Gloucestershire, who finished day three 17 runs behind and with just four of their second-innings wickets remaining.
Gareth Roderick provided some resistance with 70 but Mark Footitt has added three wickets to the three he took in the first innings to put Derbyshire on the brink of victory.
David Willey celebrated his England call-up with a destructive 88 as some solid lower-order scoring saw Northamptonshire complete their second innings on 429.
Charlie Shreck took 4-91 for Leicestershire, who were 44-1 at the close of play on day three, and require a further 332 runs on the final day at Grace Road.