In a large mixing bowl, whisk the active dry yeast into the warm water. Ensure the water is not too hot as it can kill the yeast.

Allow the yeast mixture to sit undisturbed for about 5 minutes. This is to activate the yeast, it should become frothy, indicating it's ready for use.

Add the salt, brown sugar, and melted butter to the yeast mixture. Whisk these ingredients until well combined. The mixture should be uniform and the brown sugar mostly dissolved.

Gradually add 3 cups of the all-purpose flour to the mixture, incorporating roughly 1 cup at a time. Mix well after each addition to ensure the flour is thoroughly combined.

Continue adding the remaining 1 cup of flour until the dough becomes thick and is no longer sticky. You may not need the entire amount, or may need a little more, depending on the humidity and the exact properties of your flour.

Turn the dough out onto a lightly floured surface. Knead the dough for about 3 minutes, until it becomes smooth and elastic.

Form the dough into a ball and place it in a greased bowl. Cover it with a clean kitchen towel.

Let the dough rise in a warm place for about 10 minutes. It should be noticeably puffy after this time.

While the dough is rising, preheat your oven to 400°F (200°C) and line two large baking pans with parchment paper.

After the dough has rested, divide it into 8 equal sections.

Roll each section into a long rope, about 20 inches long. Then shape each rope into a pretzel shape.

In a large pot, combine 9 cups of water and the baking soda, and bring the mixture to a boil.

Carefully place each pretzel one at a time in the boiling water and boil for 20 seconds. Use a slotted spoon or spatula to remove the pretzel and place it on the prepared pans.

Brush each pretzel with the beaten egg white and sprinkle with coarse salt.

Bake the pretzels for approximately 18 minutes, or until they are golden brown.