How do I fill a wide gap between tile and other flooring / sill plate?
Would a sill extension work for filling the gap at the entry door?
Here's the first link I found:
http://www.thebuilderssupply.com/Sill-Extensions_c_351.html

And, since you mention that its appearance is secondary, I'll throw out that you might be OK just filling that 1/2" gap with grout.  On my first tile job I didn't notice that the edge of my fiberglass tub had a slight belly, and ended up with a gap that went from 1/4" at the middle of the tub (same as my grout spacing), to about 1/2" at the head and foot corners.  I filled the gap with grout and it looked surprisingly good.  When we moved after about 4 years there was no evidence of any cracking or separation between the tub and the grout; it looked pretty much the same as when I first installed it.

EDIT: Another possibility would be to fill the gap with a length of aluminum bar stock to match the threshold.  You could remove the threshold and fasten the bar to its tile-facing edge with countersunk flat head machine screws and nylon-insert locknuts so that the screw heads are hidden in between the tile and the bar.  Assuming the cement board is butted right up to the threshold, you'd probably need a 3/8" square bar mounted high on the threshold so it overlaps the cement board.