Cheeseburger Fried Rice

SauceStep 1Bring ½ small white onion, finely chopped, 1 Tbsp. apple cider vinegar, 1 Tbsp. sugar, ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt, and 1 Tbsp. water to a simmer in a small saucepan over medium heat. Cook until onion is softened, about 1 minute. Scrape into a blender; add ⅓ cup mayonnaise, ¼ cup ketchup, 1 tsp. soy sauce, 1 tsp. yellow mustard, ½ tsp. garlic powder, and ½ tsp. paprika and blend until smooth. Transfer burger sauce to an airtight container; chill until ready to use.Do Ahead: Sauce can be made 4 days ahead. Keep chilled.Fried Rice and AssemblyStep 2Whisk 2 large eggs, a pinch of kosher salt, and a pinch of freshly ground pepper in a medium bowl to combine; set aside.Step 3Toast 1 Tbsp. sesame seeds in a medium nonstick or cast-iron skillet over low heat, stirring often, until golden, about 3 minutes; transfer to a plate.Step 4Heat 1 Tbsp. vegetable oil in same pan over medium-high. Add 8 oz. ground beef and press into a thin layer; season lightly with salt and pepper. Cook, undisturbed, until deeply browned underneath, 6–8 minutes. Add 1 tsp. sugar and 1 Tbsp. soy sauce and cook, stirring and breaking up, until meat is cooked through, about 1 minute. Transfer to another plate.Step 5Cook ½ small white onion, finely chopped, and remaining 2 Tbsp. vegetable oil in same pan, stirring often, until onion is translucent, about 1 minute. Pour in reserved egg mixture, swirling pan to coat. Cook, undisturbed, 5 seconds, then stir in a circular motion with a heatproof rubber spatula or wooden spoon until curds are just set, about 30 seconds more.Step 6Return beef to pan and add 2 cups chilled cooked short-grain white rice. Cook, stirring often, until rice is softened and heated through, about 5 minutes. Mix in 1 cup (packed) 1"-pieces iceberg lettuce and remaining 1 Tbsp. soy sauce; season with salt and pepper. Cook, stirring, until lettuce starts to wilt, about 30 seconds. Drizzle about 2 Tbsp. burger sauce over rice (save remaining sauce for another use). Arrange 2 slices American cheese on top in a single layer. Reduce heat to low, cover, and cook until cheese is melted, about 1 minute. (Or to recreate the restaurant presentation, spoon fried rice onto an ovenproof platter; top with sauce and cheese and broil until cheese is melted, up to 2 minutes.) Top with sesame seeds.Do Ahead: Ground beef can be cooked 2 days ahead. Transfer to an airtight container; cover and chill.