I had an idea for qualifying which probably wouldn't work or would require a bit of thinking/tinkering to make it work.
Session 1, a 15/20minute session like q1 now with all cars able to compete - positions 11 to 20 are set with the fastest 10 going into session 2.
Session 2 is an elimination qualifying race. Sort of like the elimination race in track cycling but instead of being the last over the line being knocked out it's the slowest driver on each lap who is knocked out. So the cars go out in order with a pre-set gap between them (fastest from q1 chooses where to start from 1-10 with the rest following in order, i.e. they can choose to go out 5th so first out would be 6th from q1, or choose to go out last which would put 2nd from q1 out first...etc or maybe that's in reverse order so they can't play games). So they fuel up for 17/18 laps and after they've all completed a lap the slowest on that lap is knocked out, then a cool down lap to charge batteries and create space while the slowest car pits, then they go again. You could get a slightly mixed order in the top 10 if someone under-fuels or uses all their electrical energy to get a higher spot or if someone makes a mistake on their lap, so:
Lap 1, out lap
Lap 2, 10th set
Lap 3, cool down
Lap 4, 9th set
Lap 5, cool down
...
Lap 16, 3rd set
Lap 17, cool down
Lap 18, is a pole position shoot out with the 2 remaining cars
It would require tyres capable of doing 20 odd laps without suffering like the Pirelli's do after 3... It's also not a format for the purists. Could be exciting, but equally could get a bit tactical.