In a small bowl, combine the warm milk, yeast, and 1 tablespoon sugar. Set aside and allow the yeast to proof for 5 minutes until the mixture is frothy.
In a large mixing bowl, combine 1 cup of the flour, the brown sugar, and salt. Add the yeast mixture and stir well.
Cut the two tablespoons of softened butter into small chunks and stir in to the pretzel batter (some small bits of butter may not fully mix in, but they will be incorporated during kneading).
Add the remaining 1 ¼ cups flour and mix well.
Turn the dough out onto a floured surface and knead for 5 minutes, adding up to ½ cup more flour if needed, until the dough is soft and less sticky (the dough should still be slightly sticky).
Shape the dough into a ball and place in a greased bowl, turning once to grease the top of the dough ball. Cover the bowl with a clean kitchen towel and place in a warm location to rise until doubled, 60-90 minutes.
Preheat the oven to 450 degrees Fahrenheit.
Prepare a large baking sheet by greasing with non-stick cooking spray or covering with a silicone baking mat.
Punch the dough down and divide it into 6 equal pieces. Use the palm of your hand to roll the dough starting in the center and working your way outward to form a long, thin rope of dough. The rope should be about 30 inches long.
In a medium bowl, combine the 3 cups warm water and 1/3 cup baking soda. Stir well.
Twist one of the dough ropes into the shape of a pretzel. Dip the pretzel into the baking soda mixture, then place on the prepared baking sheet. Repeat with the remaining ropes of dough.
Sprinkle the pretzels with coarse salt, then bake at 450 degrees for 10 minutes until golden brown. Note: the bottoms of the pretzels will get darker faster than the tops. To prevent burning, you may want to check the pretzels after 8 minutes of baking.
Remove the pretzels from the oven. Use tongs to dip the hot pretzels into the ½ cup of melted butter, or use a pastry brush to baste the tops and bottoms of each pretzel with the butter. Sprinkle with additional coarse salt if desired.
Serve the pretzels warm with dipping sauces if desired.