Set a toilet flange before laying tile, overcompensated by almost 3/4". I cut some plywood to wedge under it, and my only thought is to run a very thick bead of caulk in the gap and hope I make it look halfway decent. Has anyone ran into this issue before, and is there anything else I can use to fill this gap and make it look good? 
Toilet Flange – Fixing a Toilet That Is Too High
flangetoilet
Related Solutions
We had a similar problem with our Duravit toilet. We followed the instructions above, and after pulling out both the intake water controller portion, as well as the flush control, he didn't find any calcium build-up: just some rust stains. 
As a final check, my husband reached down into the toilet tank and felt something stuck in the bottom stopper valve. He felt what he thought was a broken piece of rubber, and after tugging on it with some force, he pulled the foreign object out in his hand from the access hole in the wall, and found a perfectly intact, but dead scorpion
]2 It must have died just that morning when the toilet started running. Luckily he wasn't stung.
For Future Viewers:
Right idea, but WAY too complicated for minimal support & no screwing for the flange. With NO FLANGE in place, build a frame first, not piece by piece in the floor. With a Flange in place you'd build it without the right side's {bottom} joist attachment so you can dry-fit everything & finish inner attachments to then LASTLY screw to the joists.
Stand up 2x4's to be attached to joists LAST. Then, bridge the Stand-Ups with 2x4's On-Flat. You screw through the Stand-Ups & into the ends of the On-Flats (do just the left Stand-Up {top} configuration if a Flange is in place).
Then, you bridge the On-Flats with a shorter On-Flat. First, screw that in from the left Stand-Up to hold it in place & toe-nail screws on either end at the Flange side. Also, toe nail what can fit On-Flat on the right side {bottom} of the Flange Finish by just screwing outward (& angled upward for this Askers I-beam pictures) through the Stand-Ups & into the joists.
This should be snug to the Flange's underside collar or the drain pipe, depending on your subfloor elevation. The On-Flats even allow you to cut or drill the platform frame for an actual tight fit complete circle. Of course, adjust this proven design to your actual application.
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Best Answer
One solution is to trace the toilet base on a piece of 3/4 plywood and carefully cut the shape out. Place the plywood with shape removed around the flange mimicking the outline of the toilet.
When you have it placed in exact position secure the cutout to the floor with tape and fill the area with sanded grout. When dry remove plywood form, set toilet on the bed of grout.
Mix a bit more of the grout and dress the edges with this. Neatly sponge the area and you should have a decent looking durable base under the toilet that will last a long time .