1. Combine first 3 ingredients in a large bowl. Let stand 5 minutes or until bubbly. Weigh or lightly spoon 5.63 ounces (about 1 1/4 cups) flour into dry measuring cups; level with a knife. Combine 5.63 ounces flour and salt. Add sour cream and egg to yeast mixture; stir until smooth. Add flour mixture; stir until a moist dough forms.
2. Turn dough out onto a lightly floured surface. Knead until smooth and elastic (about 3 minutes); add enough of remaining 1.13 ounces flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel slightly sticky). Place dough in a clean bowl coated with cooking spray. Cover dough with plastic wrap. Let rise in a warm place (85°), free from drafts, 1 hour or until almost doubled in size.
3. Punch dough down. Divide dough into 36 equal portions; roll each portion into a ball. Cover dough with plastic wrap coated with cooking spray; let stand 30 minutes.
4. Clip a candy/fry thermometer onto the side of a Dutch oven; add oil to pan. Heat oil to 375°. Combine powdered sugar, syrup, and 2 tablespoons water; stir until smooth. Place 9 dough balls in hot oil; fry 2 minutes or until golden and done, turning as necessary. Make sure the oil temperature remains at 375°. Remove doughnut holes from pan; drain. Dip doughnut holes into syrup mixture; remove with a slotted spoon. Drain on a cooling rack over a baking sheet. Repeat procedure 4 times with remaining dough balls and syrup mixture.