Best Ever Breadsticks

Combine 1-1/2 cups flour, yeast, sugar and salt. In a small saucepan, heat milk, 1/4 cup water and butter to 120°-130°. Add to dry ingredients; beat on medium speed just until moistened. Stir in enough remaining flour to form a stiff dough., Turn dough onto a lightly floured surface; knead until smooth and elastic, 6-8 minutes. Place in a greased bowl, turning once to grease top. Cover and let rise in a warm place until doubled, about 30 minutes., Punch down dough. Pinch off golf ball-sized pieces. On a lightly floured surface, shape each into a 6-in. rope. Place on greased baking sheets 1 in. apart. Cover and let rise for 15 minutes., Preheat oven to 400°. Beat egg white and remaining water; brush over breadsticks. Sprinkle with coarse salt. Bake until golden, about 10 minutes. Remove from pans to wire racks to cool.