Bring water to boil in large saucepan. Add the salt and while stirring, add the polenta. Bring back to boil, lower heat to maintain a simmer, cover and cook for about 15 minutes, stirring now and then, until soft textured. Remove from heat, stir in olive oil, butter, and cheese, cover and let sit while you cook the eggs.
Wilt the spinach in a skillet or just add it on top of the cooked polenta, cover, and let it rest while you cook the eggs.
Polenta will stay creamy in the cooking pot, covered and off heat, for about an hour. If you have leftovers, pour them into a mini sheet pan or other dish and refrigerate. Slice out pieces and warm in a frying pan or on a griddle.
Put a dab of butter or oil in each cup, and float them in a pan of simmering water (big enough pot to hold 4 at once) until butter is melted. Remove from heat, crack one egg into each cup, and return them to the simmering water. I like my eggs custardy, which I've timed to 7 minutes with the lid on the pot. Start checking at 6 minutes so as not to overcook the yolks.