1. Bring the water and salt to a boil over high heat. Once boiling, add the polenta in a slow, steady stream, whisking the entire time. Continue whisking for about a minute. Reduce heat to maintain a simmer (low to medium-low depending on your stove) and cover. Cook, whisking thoroughly every 5 minutes or so, being sure to scrape around the bottom of the pan each time so it doesn’t stick. If the polenta begins to get too thick, add another ¼-½ cup of water. Cook until polenta is soft, about 20 minutes, and remove from the heat.
2. While the water comes to a boil, remove the leaves from the chard stems and rinse both the leaves and stems to remove any grit. Trim the ends off the stems and then chop. Once the polenta has been added to the boiling water, heat a large skillet over medium heat. Once hot, add the olive oil, swirl the pan to coat, and add the chard stems and onion. Sauté, stirring frequently, until soft, 8-10 minutes. Tear or chop the chard leaves into bite-size pieces and set aside.
3. While the onion and chard stems are cooking, heat another large skillet over medium heat. Once hot, add 1 tablespoon of butter and swirl to melt the butter and coat the pan. Carefully crack the eggs into the pan and season with salt and pepper. Cook until the whites are just set. Add one tablespoon of water to the pan and cover to finish cooking the eggs. Cook covered for 15-45 seconds for runny yolks and 1-3 minutes for fully cooked yolks. Uncover and remove from the heat.
4. Once the onion and chard stems are tender, add the greens and cover and cook until they are wilted, about 4 minutes. If the pan starts to dry out and the chard starts to stick to the pan, add another tablespoon or two of water.
5. Add the remaining 2 tablespoons of butter and ¼ cup of cheese to the cooked polenta. Stir well before tasting and seasoning with salt and pepper as desired.
6. To serve, divide the polenta among four plates or wide bowls. Top each with a quarter of the greens and two eggs. Sprinkle the remaining cheese over each dish and serve immediately.