Lee carded the equal-low second round, a three-under-par 67, to head up the Australian challenge five shots off the pace in Rochester.
The 24-year-old had the round of the day going after collecting five birdies in his first seven holes before his putter cooled off on a freezing cold Friday at the season's second major.
Lee dropped a shot at the brutal par-4 sixth - ranking this week as the most difficult major championship hole in the past 20 years - before missing a six-foot birdie attempt at the 13th and three-putting on No.15 for bogey.
Still, after clawing his way back to even with the card, Lee was tied for 10th behind joint leaders Corey Conners and Scottie Scheffler, who both shot 68s.
Cam Davis is the next best Australian, six back at one over, after a second-round 70, with Adam Scott a further stroke back and needing to play catch-up over the weekend to move back into contention after another disappointing finish to his second round.
Scott tumbled down the leaderboard from a share of sixth at the start of the round after leaking four shots in his last four holes in his four-over round of 74.
Scott had been cruising in red numbers for most of the day until his late stumbles.
He also undid some great work on Thursday with a double-bogey six at the last hole, having shared the lead during the first round.
Lucas Herbert, the only Australian to par the sixth, rebounded from a first-round 75 with a 69 to make the halfway cut at four over.
Big guns Cameron Smith and Jason Day were still battling through their second rounds, desperately trying to beat the projected cut line of five over.
British Open champion Smith was at four over through 14 holes, while Day birdied the 12th and 13th holes to pull himself back to six over.
David Micheluzzi, with rounds of 76-73, missed the cut on his major championship debut.
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