The first stoppage happened when Aspar's Hector Faubel had a very violent high-speed at the final corner. Faubel escaped serious injury but has been taken to hospital for checks due to chest and back pain.
Kent then led much of the restarted session, but would not secure his maiden Moto3 pole as Cortese came through to depose him by just 0.083 seconds.
The duo's final pole shots were then halted when Alex Marquez had a similar accident to Faubel, causing a second red flag with 23 seconds left on the clock. It was Marquez's third fall of qualifying.
The stoppage interrupted many riders' final laps, but no one was quite as frustrated as Laglisse rider Efren Vazquez, who set a quicker time than Cortese moments after the red was shown, meaning it would not count. Rather than starting from pole, Vazquez will start seventh.
Cortese's title rival Maverick Vinales got himself onto the front row in third at the very end, with Luis Salom, Zulfahmi Zhairudden and Alex Rins filling row two.
Among the many other incidents in the session, Jack Miller sustained another broken collarbone in a fall. His bike then slid into Adrian Martin, taking the Laglisse rider down as well.
In Moto2 Pol Espargaro beat title rival Marc Marquez to pole position at Indianapolis.
Unlike the preceding Moto3 and MotoGP session, Moto2 qualifying saw little incident and a relatively static lead order. Pons rider Espargaro established provisional pole with a 1:42.602 early on and was able to sit on a 0.231-second cushion back to Marquez, who he trails by 34 points in the standings. Marquez's late runs were spoiled by traffic and mistakes.
Fellow championship contender Andrea Iannone hauled himself up to the final front row position in the closing stages, having previously been outside the top 10.
Julian Simon qualified fourth, sharing row two with Dominique Aegerter and Scott Redding.
There was no repeat of the violent accidents that had marred and delayed Moto3 and MotoGP qualifying. In one of the few incidents, Alex de Angelis fell early on, leaving him 16th on the grid.