Melt 3 tablespoons of the butter in a large pot over medium heat until foamy. Add half of the greens, and use tongs to turn them over a few times until they are coated with butter and start to wilt. Add the remaining greens and repeat--the greens will be up to the top of the pot, most likely, but they'll cook down. Add 1 3/4 cups water, bring to a boil and cover. Reduce the heat to medium-low, and simmer, turning occasionally, until dark green and crisp-tender, about 15 minutes.
When the greens have simmered about 10 minutes, heat a medium nonstick skillet over medium heat. Add the remaining tablespoon butter. When the foam begins to subside, crack the eggs into the skillet. Cook until the edges are crisp but the yolks still runny, about 3 minutes.
Put the miso in a small bowl. When the greens are done, ladle some of the cooking water (also known as pot liquor) into the miso. Whisk until smooth then stir the miso back into the greens, cover, and set aside until the eggs are ready.
Use tongs to divide the greens among four bowls, then ladle some of the broth over them. Top each one with a fried egg. Pass the hot sauce so that everyone can splash as much they want over their greens.
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