Is it possible to avoid anaerobic 'burn' on ascents? As a total noob to cycling the thing that really kills me is hills steep enough I get into anaerobic territory. My general fitness lets me cycle for hours without really getting tired even working fairly hard, but my thighs have a pretty limited total amount of ascent in them which takes a long time to replenish.

I feel like on steeper hills (say 10% and above) I have to go anaerobic to get enough speed... if I shift down to my lowest gear at my regular cadence I am so slow I risk falling off, and of course the climb just takes much much longer anyway!

My beginner question is: should it be possible to get up most/all hills aerobically using gears, or is it a fundamental feature of cycling that you will end up burning, and you will find sometimes your legs just give out mid-hill until you improve your muscles/fitness?
Get an app like strava to log your rides, and see your progress on those climbs.

I have a climb near home which took three hours first time I did it.  I rested a bunch of times on the way up, and got passed by dozens of riders.   

Over the last three years I've improved to the point I can do the climb in 40 minutes non-stop, still get passed but I'm passing others, some of whom look shattered.

Remember it never gets easier, you just go faster.  Some logging will help show you your improvements, because it never feels faster at the time.