Combine bread flour (3 cups/375g), brown sugar, yeast, and salt in the bowl of a stand mixer that’s been fitted with a dough hook. Stir until combined.
Add melted butter and water and stir on low speed until completely combined. Continue to stir until a smooth elastic dough forms that clings to itself and pulls away from the sides of the bowl. Dough should be slightly tacky but not unmanageably sticky. If needed, stir in additional bread flour, 1 Tablespoon at a time (I often need to add 1-2 additional Tablespoons).
Lightly brush a large bowl with olive oil and place your dough ball into oiled bowl. Cover tightly with plastic wrap and then place in a warm place to rise for 30 minutes/until doubled in size (the top of my oven is usually quite warm while preheating, so I’ll usually turn on my oven to 450F/230C at this point and place my dough on top).
Preheat oven to 450F/230C and line 2 baking sheets with parchment paper. Set aside.
Once pretzel dough has finished rising, transfer dough to a clean, lightly floured surface and divide dough into 10 even pieces.
Roll each piece of dough into a long thin rope (mine were about 20-24” long!) and twist into a pretzel shape (see photos in post [above] or demonstration in video [below] for example). It's easy to roll the dough between your palms while stretching. You may also roll it with your palms against a clean surface, don't flour the surface very much or it will be difficult to work the dough.
Combine water and baking soda in a large deep dish (I like to use a Dutch oven) and bring to a boil.
Once boiling, gently lower each pretzel into the boiling water/baking soda mixture. Boil for 30 seconds on each side then drain and place on prepared baking sheet. Repeat with remaining pretzels, spacing at least 1 ½” apart.
Whisk together lightly beaten egg and 1 Tablespoon water and brush over each pretzel. Sprinkle with coarse sea salt or pretzel salt, as desired (see notes for instructions on making cinnamon sugar soft pretzels).
Transfer to 450F (230C) oven and bake for 9-12 minutes or until golden brown.
Remove from oven, allow to cool several minutes, then enjoy. Pretzels are best enjoyed warm the day they are made.