I had a bit of the same problems you did the first time I made it, here
It may be because you used a different type of egg plant, but because I had a bit of the same issues of tough skin I cooked it at a little bit lower temperature the next time I made it and that solved it. I used the full amount of oil, but didn’t have any of the splatter issues you’re talking about. I used my 12 inch every day pan. Measured the oil in a Pyrex measuring cup, cooked the diced eggplant in the pan, used a the slotted spoon to move the egg plant to a bowl, poured off the extra oil back into the Pyrex, put everything back into the pan and finished. I found you can do this much earlier in the day or ahead of time at some point and reheat it. This cuts down to only doing the polenta at the last minute but you really don’t even have to do that at the last minute as this dish reheats beautifully, though I reheat each part separately.
The most challenging part of this recipe to me is getting the corn off the cob, but the fresh polenta is so worth it.
In the end my dirty dishes are as follows:
For eggplant: Pyrex measuring cup for oil, 12 inch pan, slotted spoon, Bowl for egg plant which could be a serving bowl later if you want one.
For corn: large stove top pan, bigger Pyrex measuring cup to drain liquid, food processor, slotted spoon used from above. (I may forgo saving the drained corn liquid as I haven’t once needed any.)
Prep cutting board/knife/measuring spoon/small spatula for tomato paste, And since I use a larger can of diced tomatoes measured those in a pyrex cup.
And yes the polenta turns to glue, so I rinse right away as I go.
Glad you liked it! I’ve made it several times now as a make ahead dinner and warmed each separately in the microwave resulting in dinner in less than five minutes. I love that!