In a small saucepan heat and stir milk, 1/4 cup water, and sugar over low heat until warm (110°F to 115°F). Pour into the large bowl of an electric mixer. Sprinkle with yeast; let stand about 5 minutes or until foamy. Add 2 1/3 cups of the flour, the butter, and 1 1/4 teaspoons salt. Using the hook attachment, beat on low speed for 2 to 3 minutes or until combined, scraping sides of bowl occasionally. Beat on medium-low speed for 8 to 10 minutes or until a soft dough forms. (The dough should pull away from sides of bowl but still be slightly sticky. If it is too sticky, beat in the additional flour.)

Turn dough out onto a clean, dry work surface. Knead a few times, then shape dough into a ball. Place in a lightly greased bowl, turning once to grease surface of dough. Cover and let rise in a warm place until double in size (about 1 hour).

Preheat oven to 450°F. Line a large baking sheet with parchment paper; set aside.

Punch dough down. Turn out onto a clean, dry work surface. Divide dough into eight portions. Roll and stretch each portion into a 30-inch rope. If the dough seems tight and does not want to stretch, cover and let rest about 10 minutes or until it relaxes.

To shape each pretzel, hold one end of a rope in each hand and form a U-shape. Cross the ends over each other twice. Then lift the ends across to the bottom of the U-shape; press to seal.

In a deep dish or pot slightly larger than each pretzel stir together 3 cups hot water and baking soda. Using a slotted spoon, lower pretzels, one at a time, into baking soda mixture for 10 seconds. Remove pretzels and place on paper towels to blot the bottoms dry. Arrange 2 inches apart on the prepared baking sheet.

In a small bowl combine egg yolk and the 1 tablespoon water. Brush pretzels with egg yolk mixture. If desired, sprinkle with coarse salt. Bake for 10 to 12 minutes or until deep golden brown. Cool slightly on a wire rack; serve warm.