I left my veggie stew burger mixture in the refrigerator for a couple hours and then finally got down to bake them. I am really pleased with the results. They cooked all the way in the correct amount of time and actually will hold together.
I used wheat flour because that is what I had around but I probably won't do that again. It made a strong wheat flavor which isn't bad but just not what I want in a veggie burger. I will either try working with chickpea flour (never have so I don't know the flavor), use oatmeal, or I will substitute beans instead so it will be more like the black bean burger that I did a few weeks ago.
I am guessing that over-processing the mixture was a huge part of my failure the first time around. I will definitely keep working at this.