In a small bowl, combine warm milk, yeast, and 1 tablespoon of sugar. Let the mixture sit for 5 minutes to proof until it becomes frothy.
In a large mixing bowl, combine 1 cup of flour, brown sugar, and salt. Add the yeast mixture and stir until well combined.
Cut the softened butter into small chunks and stir it into the dough mixture (some bits of butter might not fully incorporate, but they’ll blend in while kneading). Add the remaining 1 ¼ cups of flour and mix until fully combined.
Turn the dough out onto a floured surface and knead for 5 minutes, adding up to ½ cup more flour if necessary, until the dough is soft and slightly sticky.
Shape the dough into a ball and place it in a greased bowl, turning it once to coat the top. Cover with a clean kitchen towel and let it rise in a warm spot for 60-90 minutes, until doubled in size.
Preheat your oven to 450°F and prepare a large baking sheet by greasing it with non-stick spray or covering it with a silicone baking mat. Punch down the dough and divide it into 6 equal pieces. Roll each piece into a long, thin rope, about 30 inches in length.
In a medium bowl, combine the warm water and baking soda, stirring well.
Shape one dough rope into a pretzel, dip it into the baking soda mixture, and place it on the prepared baking sheet. Repeat this for the remaining dough ropes.
Sprinkle each pretzel with coarse salt and bake for 10 minutes, or until golden brown (check at 8 minutes to prevent burning as the bottoms may brown faster).
Remove the pretzels from the oven. Use tongs or a pastry brush to dip or baste the pretzels with melted butter.
Optional: sprinkle with more coarse salt. Serve warm with dipping sauces, if desired.