Heat milk in a small sauce pan over medium heat (or in a microwave) to 110°-115° F.
Transfer to a 2-cup measuring cup, stir in 1 tablespoon of the sugar and whisk in the yeast.
Let it sit for about 5 minutes or until the yeast becomes foamy.
Then whisk in the two eggs.
Combine the remaining sugar, flour, and salt in the bowl of an upright mixer.
Add the yeast mixture and with the mixer running, add the butter one piece at a time.
Mix on medium speed until dough ball forms and the ingredients are well blended.
Then knead on medium-high speed for 5 minutes, or until the dough is soft and smooth.
Brush a medium bowl with melted butter, place the dough in the bowl, and brush the top of the dough with melted butter.
Cover with plastic wrap and allow dough to double in size (about 1 1/2 hours).
To make the pretzels, punch the dough down and turn onto a flour-dusted surface.
Cut the dough into 12 equally-sized pieces.
While shaping the pretzels, keep both the pieces of unshaped dough as well as the shaped pretzels under a dish towel to prevent drying.
Roll each piece into a rope about 1 1/2 feet long, and shape into a pretzel knot.
Place the shaped pretzels on two parchment paper-covered baking sheets.
Cover with a dish towel and allow the pretzels to become puffy. About 30 minutes. They do not have to double in size.
While the pretzels are rising, preheat the oven to 375° F and place one rack in the top third of the oven and the other in the bottom third.
When the pretzels are puffy, whisk the water and the egg white together.
Brush the tops of the pretzels with the egg wash, then sprinkle with pearl or turbinado sugar.
Place the baking sheets in the oven and bake for 10 minutes.
Rotate the baking sheets in the oven and bake until they are golden brown, about 6-8 minutes more.
Serve warm or at room temperature.