In skillet, heat butter. Add onion; cook over medium heat 5 minutes. Reduce heat to low. Sprinkle with 2 teaspoons sugar; cook 20 minutes, until onions are very soft and lightly browned. Remove from heat.
In bowl, combine 1 teaspoon sugar, the water and yeast. Let stand 5 minutes, until foamy, to dissolve yeast.
In medium-size bowl, mix 2 cups flour, salt and remaining sugar.
Stir buttermilk, butter, onion and flour mixture into yeast mixture. Stir in 1-1/4 cups flour. Knead until smooth and elastic, 5 minutes, adding flour as needed to prevent sticking.
Place dough in large greased bowl; turn dough to coat. Cover with clean kitchen towel; let rise in warm place until doubled in volume, 1 hour.
Punch down dough. Let rest 5 minutes. Coat 2 large baking sheets with cooking spray. Divide dough into 6 equal pieces. Roll 1 piece into 20-inch long rope. Cut rope into 4 equal pieces. Roll each piece into 8-inch-long rope. Tie each into knot. Transfer knots to baking sheet. Repeat process for a total of 24 rolls.
Cover with clean kitchen towels. Let rolls rise in warm place 45 minutes or until doubled in volume.
Heat oven to 375°F. Uncover pans. Brush risen rolls with egg white.
Bake in 375°F oven 8 minutes. Rotate pans. Bake 10 to 12 minutes, until rolls are lightly browned. Transfer rolls to wire rack; let cool slightly. If baking ahead, warm before serving.