Here's some ideas, preceded with some "hope it's not too late" cautions:
If the EZ-out is in there crooked or off axis so it's cutting into the side wall of the original hole - that's a very bad thing.
EZ-outs, thread taps as well, can be bad news in small sizes. They are hard and cheaper ones can be overly brittle. Go gently and easy, you can feel the torque and stop in time with practice. Time is your friend, apply penetrating oil and wait some or tap it with small hammer to break up any corrosion. Apply heat if you can.
When initially experiencing driver bit slipping, camming or "rounding out" the screw. Stop!! Check for warn tool bit. Apply penetrating oil. The very best penetrating oil is a home brew, 50/50 mixture of ATF (transmission fluid) and acetone - shake before use if it separates.
You can try heating the fastener if materials are "safe" to heat.
A dab of valve lapping compound applied to screwdriver tip or fastener will enhance friction and help prevent camming.
Superglue will sometimes work between driver tip and fastener sort of like the lapping compound - delay turning for a minute or two. Obviously don't get the glue down into the threads, best to apply a tiny bit to
Depending on fastener size, for bigger ones, they make a special fastener removal welding rod. You place a nut over the hole and strike an arc down in the hole, moving the rod out. The flux protects the original hole threads and the weld puddle is then "spread" into the nut. Apply wrench and out it comes - - usually. If even a small fastener is broken off at or above the surface, a small TIG welder can be used to weld on a small nut - works the same as the special welding rod, only with more finesse.
Use a rotary tool (Dremel etc) and small abrasive cut-off wheel to cut a new screwdriver slot in the fastener. In this case you would want to grind down the wheel to a smaller diameter first, so there's less collateral damage. This obviously works best on a fastener that is not broken off deep.