Shishito Pepper Potato Hash with Fried Eggs

In a large pot fitted with a steamer basket, steam the potatoes until tender, about 20 minutes. Transfer potatoes to a large bowl until cool enough to handle. Using the palms of your hands, gently smash the potatoes. Add the shishitos, cheese, scallions, and 3/4 teaspoon salt to the bowl and, using a wooden spoon, mix gently to combine. In a large nonstick skillet over medium heat, melt 1 tablespoon butter. Add potato mixture and fry, flipping once or twice, until cheese is melted, peppers are soft, and mixture is golden brown, about 10 minutes. Remove from pan and divide among four plates. Wipe out the skillet and heat the remaining 2 tablespoons butter over medium-high heat. Crack in the eggs and fry to desired doneness. Top hash with fried eggs and serve immediately with greens and hot sauce.