One caution about using Easy-Outs: you definitely don't want to wind up
with a broken-off Easy-Out stuck in the fastener that you're trying to
remove!  They are quite tough, and you can't attack a broken Easy-Out
with a drill.  The advice of using the largest one possible should be
followed.

I've seen a technique for removing broken bolts (especially with a
stuck Easy-Out)  which involves using a core drill whose O.D. is the
same as the minor diameter of the broken bolt.  A drill press (with a
magnetic base, if the bolt is stuck in cast iron or steel) is used to
core out the bolt.

The core drill is made from a piece of drill rod which is chucked in a
lathe, drilled out to the necessary depth, then teeth are filed on the
business end and the piece is hardened and tempered.

There's no reason that you couldn't use the same technique with wood.
As an earlier note suggested, when plugging the hole, use a plug
cutter to make a cross-grain plug so the new screw will grip well.

I once salvaged an engine block which had a broken-off tap stuck in a
hole.  The tap had not just broken off; it had split lengthwise!  It
was impossible to remove, because the longitudinal split caused the
remains of the tap to wedge when attempting to turn it.

I was able to get a dentist's diamond drill point and, with that
chucked in a MotoTool's handpiece, I cut the tap into pieces and
removed it in just a few minutes.  Now I always have a few of these
lying around, and they are just great for oddball jobs like this.
Look for dental supply houses, or ask your dentist where he buys
them.  I got mine from Colwell in Champaign, Illinois.