Cheddar Rice Fritters

Bring water and milk to a full boil in a 2- to 2 1/2-quart heavy saucepan. Stir in rice, then cover with a tight-fitting lid and reduce heat to low. Cook rice, undisturbed, until liquid is absorbed and rice is tender, about 15 minutes. Remove from heat and let stand, covered, 5 minutes. Fluff rice with a fork, then transfer to a large bowl and stir in cheese, butter, scallions, 3/4 teaspoon salt, and 1/4 teaspoon pepper until cheese is melted. Cool, stirring occasionally, 15 minutes, then stir in eggs until combined well. Stir together bread crumbs and remaining 1/2 teaspoon salt and 1/4 teaspoon pepper in another large bowl.
Working in small batches, drop rounded tablespoonfuls of rice mixture into bread crumbs, rolling to coat and knocking off excess, then transfer to a tray. Heat 1 1/2 inches oil in a 4-quart heavy pot over moderately high heat until it registers 365°F on thermometer, then fry fritters in 4 batches, stirring occasionally, until golden, about 3 minutes per batch. Transfer with a slotted spoon to paper towels to drain. Return oil to 365°F between batches.