In bowl of electric mixer with paddle, stir 1 envelope yeast into 1/2 cup plus 2 tablespoons warm water. Let stand until foamy, about 5 minutes. Add 1 cup flour; beat with paddle on low until combined. Cover; chill this "sponge" overnight.
Remove sponge from refrigerator; bring to room temperature. In separate bowl, mix remaining envelope yeast, 2 tablespoons warm water, and the buttermilk. Let stand until foamy, about 5 minutes.
Mix remaining 2 1/4 cups flour, sugar, and salt. Add buttermilk mixture, egg yolks, and cooled melted butter. Beat on low until dough comes together, about 2 minutes. Add sponge; beat until dough is soft and sticky, 3 to 4 minutes. On clean surface, knead a few turns into a ball. Coat bowl with cooking spray; add dough smooth side up. Cover; let stand in warm place until doubled in bulk, about 3 hours.
Cut dough into thirds; cover with plastic wrap. On lightly floured surface, roll out each piece 1/4 inch thick. Using 2-inch and 3/4-inch cookie cutters, cut out doughnuts and centers. Place on parchment-lined baking sheet. Let stand in a warm place to rise, 15 minutes.
In deep fryer, heat oil to 350 degrees. Fry doughnuts, turning once, until golden brown, about 1 minute total. Drain on paper towels; coat with sugar while warm.
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