Heat a large skillet over medium-high heat. Once hot, add the butter and swirl it around to coat. Once the butter is bubbling and just turning light brown, add the eggs, then drizzle the soy sauce all around the eggs (there should be a lot of sizzling going on!).
Cook until whites are cooked through, reducing the heat if the soy sauce begins to burn, about 2 minutes (to get the egg whites all the way cooked through, it helps to cover it with a lid towards the end when there is just a little bit of uncooked white on top).
Turn the heat down to medium and use a spatula to transfer the eggs to a plate. Add the cooked rice and greens to the skillet and use your spatula to toss it in the buttery soy sauce mixture, scraping up anything stuck to the pan. Stir in the rice vinegar and sesame oil, continuing to scrape the bottom of the pan and mix the rice.
Serve the rice topped with the two fried eggs. Garnish with scallions, sriracha (if using), and sesame seeds (if using). Break the fried egg open and mix everything into the rice.