How do I use a PWM unit and/or microcontroller to have transistors progressively all turn on and then progressively all switch off?
Just do it in a periodic interrupt.  It appears you have up to 36 events 60 times per second, for a total event freuency of 2.76 kHz.  That's still "low" for many processors.  Even just a 10 MIPS processor will execute 3600 instructions per interrupt.  What you want to do will take nowhere near that.

You can even accomodate the 36 events per cycle being somewhat unevenly space in time.  Using the same 10 MIPS processor for example, you could set up the interrupt every 600 cycles (60 &micro;s).  That's still way more cycles than it takes to figure out what phase you're in and change the appropriate output accordingly.  But this interrupt is now 6 times faster than your events occur on average, so you have extra resolution in placing event within a cycle.  And that was only interrupting every 600 cycles on a slow processor.

Somewhat of a aside, but it sounds like you're trying to make 60 Hz power the hard way.  Are these transistors each driving a separate tap of a transformer primary?  That should work, but will be rather expensive and bulky compared to a more common solution like a class D drive with the PWM duty cycle adjusted each cycle to result in the 60 Hz sine output.