Heat the olive oil over medium-high and add the onion. Saute, stirring occasionally, until onion begins to brown on the sides, about 10 minutes. Reduce the heat to medium-low and continue cooking, stirring occasionally until onion has almost caramelized, about 20 minutes.
Add the peppers and mushrooms (increase the heat if necessary) and cook until peppers and mushrooms have softened and turn golden-brown, about 8 minutes. (Note: you can prepare the polenta while the vegetables are cooking).
Heat the water in a deep saucepan to a full boil and add the sea salt. Gradually add the polenta, stirring constantly. Add the butter and continue cooking at a simmer, stirring constantly, until polenta is thick and creamy, about 20 to 30 minutes. Taste polenta for flavor and add more sea salt and/or cheese to taste.
Serve polenta with caramelized onions, peppers, and mushrooms on top, along with a sprinkle of feta (or parmesan cheese) and chopped oregano.