Poach the eggs: Bring a wide pot of water to a boil, reduce heat to a simmer, and poach the two whole eggs for about 3 minutes and 45 seconds, or as desired. Remove the eggs to a bowl of warm water and set aside.
Beat the remaining egg white briefly with a fork. Beat the egg white briefly with a fork. Break up the leftover risotto with a spoon or fork and mix it with the egg white until well combined. The mixture should look wet but not runny, and a spoonful of rice should hold together.If the mixture seems too runny, you can add some grated Parmesan, breadcrumbs or even a little flour to help hold it together. If it’s too dry and the rice won’t stick together, beat the egg yolk and add it a little at a time until you get the right consistency.
Heat the oil over medium to medium-high heat in a large cast iron or nonstick frying pan (I use a ceramic nonstick pan). When the oil is hot, use an ice cream scoop or a large spoon to carefully drop about 1/3- to 1/2-cup scoops of risotto into the pan. Press down gently to form the mounds of risotto into disks, and press any risotto clumps that fall off back into the sides of the cakes.
Let the risotto cakes cook undisturbed until you can slide them around as complete units the pan (if the cake is breaking apart or clumps of rice are falling off when you slide it, it’s not ready yet). When the cakes move easily and the bottoms are golden brown, carefully flip them over. Here’s my method, which is a little bit like dealing with very small frittatas: Position a large spatula on one side of the cake. Place a second spatula on the opposite side of the cake and wiggle it underneath. Quickly flip the cake onto the first spatula, then use the second to slide it off back into the pan. Push any bits of rice that have fallen off into the sides of the cake.
Continue to cook until the second side is golden brown and the cakes are cooked through. Season the outsides with salt and pepper (optional, but it adds another dimension of flavor to the crispy exterior). Divide cakes between two plates and top with poached eggs.
Salad with Balsamic Vinaigrette
Combine the oil, vinegar, mustard, honey, basil, and salt and pepper to taste in a jar and shake vigorously until combined.
Divide the salad greens between the two plates holding the risotto cakes and eggs. Drizzle the dressing over everything.