In a large mixing bowl combine 2 cups of the flour, the yeast, and cinnamon; set aside. In a medium saucepan heat and stir milk, butter, sugar, and salt until warm (120 degree F to 130 degree F) and butter is almost melted.
Add milk mixture to flour mixture. Add the 3 eggs. Beat with an electric mixer on low to medium speed for 30 seconds, scraping sides of bowl constantly. Beat on high speed for 3 minutes. Using a wooden spoon, stir in apricots and as much of the remaining flour as you can.
Turn dough out onto a lightly floured surface. Knead in enough of the remaining flour to make a moderately soft dough that is smooth and elastic (3 to 5 minutes total). Shape into a ball. Place in a lightly greased bowl, turning once to grease surface. Cover and let rise in a warm place until double in size (about 1-1/2 hours).
Punch dough down. Turn out onto a lightly floured surface. Cover and let rest for 10 minutes. Lightly grease 2 baking sheets; set aside.
Divide dough into 20 portions. Shape each into a smooth ball. Place 1-1/2 inches apart on the prepared baking sheets. Cover and let rise in a warm place until nearly double in size (45 to 60 minutes).
Using a sharp knife, make a crisscross slash across top of each bun. In a small bowl combine egg white and water; brush over buns.
Bake in a 375 degree F oven for 12 to 15 minutes or until golden brown. Cool slightly on wire racks. Makes 20 buns.