After removing the starter, the piece can likely be retrieved by drilling the center of the remaining stud and using an easy-out. Another method is using a reverse rotation drill bit. The remaining piece is not going to be that tight in the block, because the missing bolt that held tension against the threads is now missing.
Getting the broke part out is the best method, no need to drill out for larger hole.
Do some searching on "starter bracket", which will lend some info onto possibly why you did break a starter bolt.
This missing bracket and or bolts/nuts that attach it will almost certainly lead to a broken starter bolt eventually.
Been there done that too 
1982 C10 SCSWB 6.5TD, mods too extensive to list. (13.69 1/4 mile @94.6 MPH) RACE TRUCK
1982 C10 SCSWB 6.2NA, 2.73 700
1986 C10 SCLWB 6.2TD 3.73 700
1989 V20 SUB 6.2NA, 3.73 400
1994 G20 VAN 6.5NA, 3.42 60E
1994 K20LD ECSWB 6.5TD, 3.42 80E
1995 K20 SUB 6.5TD, Wrecked, ran into by stupid teen.
1995 C3500HD DRW 6.5TD, 12' Flatbed 5.13 80E
1995 C3500HD DRW 6.5TD, 18' Rollback Wrecker 4.63 80E
1994 C20HD ECLWB 6.5TD 3.73 80E Wifes Truck.
1995 C20LD ECSWB 6.5TD 3.73 80E
1995 K20LD SCLWB 6.5TD 3.73 80E
1996 K30 DRW 6.5TD 4.10 80E
1997 C10 Tahoe 2Door 2WD 5.7L to 6.5 Conversion Underway
. and since i don't have a tap and dye set, i'm gunna be forced to pay someone else to touch my baby. this is a sad day for me, and my wallet. so do you guys think that after they get the broken piece out, are they doing to have to tap it for a larger bolt, like i thought i would of had to do. i know the machanic pretty well and he's gunna work with me on the price, but it still makes me really mad, exspecially that i have to miss work today.
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