1. Bring 6 cups water to a boil in a saucepan over medium-high heat. Stir in rice and 1/4 tsp. salt. Reduce heat to medium, and cook, stirring often, 25 to 30 minutes. (Rice will be very soft and thick.) Remove from heat, and drain. Place 1 1/2 cups cooked rice in a bowl, discarding remaining rice. Mash rice with a potato masher 30 seconds. Cool 20 minutes or until lukewarm.
2. Stir together warm water, yeast, and 1 tsp. granulated sugar in a 1-cup glass measuring cup; let stand 5 minutes. Stir yeast mixture into rice. Cover with plastic wrap, and let stand in a warm place (80° to 85°), free from drafts, 8 to 12 hours.
3. Stir eggs into rice mixture. Combine flour, next 2 ingredients, and remaining 1/2 tsp. salt. Stir flour mixture into rice mixture. Cover with plastic wrap, and let stand in a warm place (80° to 85°), free from drafts, 30 minutes.
4. Meanwhile, pour oil to depth of 3 inches into a deep cast-iron skillet or large Dutch oven; heat to 350°. Drop dough by rounded tablespoonfuls into hot oil, and fry, in batches, 3 minutes or until golden brown. Drain on paper towels. Sprinkle with powdered sugar, and serve immediately.
Note: We tested with Water Maid Medium-Grain Enriched Rice.