In a medium saucepan over medium heat, melt the butter. Add the chopped green onion, and cook for 2 to 3 minutes. Add the rice, and cook, stirring frequently, until rice grains are translucent, 3 to 4 minutes.
• In a medium saucepan over medium-high heat, heat broth until gently simmering. Add 1/2 cup warm broth to the rice, and stir until most of broth is absorbed. Each time rice begins to look dry, add broth, 1/2 cup at a time, stirring well and allowing each addition to be mostly absorbed before adding more. Cook, using all remaining broth, until rice is creamy and thick, 30 to 35 minutes.
• Add the pumpkin purée and cheese to rice, stirring until cheese is melted. Set aside to cool, and then refrigerate for at least 2 hours, or until cold.
• Shape risotto into 24 (1- to 1 1/2-inch) patties. Set aside.
• In a small bowl, whisk together the egg whites and water.
• In a shallow dish, combine the panko, salt, and pepper, and stir to combine.
• Dip the risotto patties in the egg mixture, then roll in panko, dusting off excess crumbs. Set aside.
• In a large skillet, add oil to a depth of 2 inches. Heat oil over medium-high heat until it reaches 375°.
• Fry patties, 3 to 4 at a time, until golden brown. Drain on a plate lined with paper towels.
• Top each with 1 to 1 1/2 teaspoons sour cream. Garnish with a green onion slice, if desired. Serve immediately.
*Arborio rice is a high-starch Italian rice. It can be found at most grocery stores and at specialty-foods shops.