Steps:

• Sprinkle the rice and salt into the boiling water.
• Cover and cook slowly until the water is absorbed, 7-10 minutes.
• Add the milk, stir, cover, and cook 10-12 minutes more, or until the rice is tender.
• Stir in 1 egg and 2 tablespoons of the butter.
• Spread the mixture on a shallow plate, cover with plastic wrap, and refrigerate.
• Beat the remaining egg in a shallow dish and put the crumbs on a piece of wax paper.
• Shape the well-chilled rice mixture into 6 conical or patty shapes, pressing firmly together.
• Carefully dip each cake into the egg and then cover with crumbs.
• Melt the remaining 4 tablespoons of butter in a skillet with the oil.
• When hot, fry the cakes until golden brown; don't cook them too quickly or the insides will remain cold.

Steps:

• In a large bowl, gently mix everything except the butter. Over medium-high heat, heat the butter in a large frying pan until it is foaming. (A nonstick pan is good for this purpose but cast iron can work well, too.)
• Working in batches if necessary, and adding more butter as needed, use a large spoon or measuring cup to place a scoop of the rice mixture in the pan. Press down with the back of a spatula to form a patty. Cook 3 to 4 minutes, or until golden and crisp on the bottom. Gently flip patty and cook another 3 minutes or until golden. Keep warm on a paper-towel-covered plate or pan in a warm oven.