Cantilevered roof overhangs
We don't know the distance from the 4x12 beam over to the next support. So, we don't know the load (tributary load in pounds per linear foot) on the 4x12 and therefore we don't know how much, if any, the 4x12 can be notched.

However, assuming you have calculated the Total Load (LL + DL) correctly at 90 psf, then a No. 1 Doug. Fir 2x8 at 24" o.c. with a load of 90 lbs. per square foot (180 plf) can span a maximum of 9'-0" with a bending stress of fb = 1500 and modulus of elasticity = 1.4.  If you need the 2x8 joist to span further, you could span 11'-0" at 16" o.c. Or span 12'-0" at 12" o.c. (This does not account for "short duration loading" (snow that lasts less than 30 days) or "impact loading" (snow that lasts less than 7 days) like where I live. 

Therefore, to determine if the 4x12 beam is adequate, I calculated the maximum tributary load on the 4x12, which is half the 9' span plus the 2' overhang, which is:
4.5' + 2' = 6.5' x 90 psf = 585 plf 
The maximum load on a 4x12 spanning 10' is about 700 plf (Therefore, the 4x12 is slightly over sized and can be notched...if the 2x8's only span 9', which I'm guessing they are over stressed.)

So, I think there are 3 options:

Option 1: The 2x8 joists are in "reverse-bending" as they pass over the 4x12 beam. So, the top half of the joist is in tension and the bottom half is in compression. If you carefully notch the bottom one-third of each 2x8 so that the joist fits tight over the 4x12, then the overhang won't sag or fail. 

Option 2: if you lower the 4x12 enough to let the 2x8 joists to sit on the 4x12, there would be no need to notch the joists.

Option 3: The 4x12 is slightly over designed, you could notch the 4x12 beam a maximum of 2" (making the beam an equivalent of a 4x10). 

Note: In your original statement you indicated that the total load is 90 psf. This seems extremely high. Did you mean 90 plf on the 2x8 joists. If so, that would make the snow load 40 psf, which is high, but reasonable. (Meanwhile, I'd double check the span of those 2x8's.)