First, warm the milk in a small saucepan until the temperature of the milk reaches 110 degrees, about 1 1/2 - 2 minutes on medium-high heat.
Pour the warm milk along with yeast into the bowl of an electric stand mixer and whisk together until the yeast has dissolved. Then let it rest for about 5 minutes. Add brown sugar, 4 tbsp softened butter, 1 cup flour, and 2 tsp fine sea salt to the milk mixture and stir until blended using the whisk attachment.
Switch the attachment to a dough hook. Add the remaining 3 1/2 cups flour and kneaded mixture on medium-low speed until elastic.
Mix in up to an additional 1/2 cup flour, as needed, until the dough pulls away from the sides of the bowl. Cover the bowl with plastic wrap and allow it to rise in a warm place until it's doubled in size, about 1 hour.
Preheat the oven to 450 degrees. Line two baking sheets with parchment paper. Punch the dough down several times to release any air pockets. Divide the dough into 12 equal pieces and cover loosely with plastic wrap. Roll each piece out into a long, thin rope about 32 to 36 inches long. Form the dough rope into a pretzel shape.
In a shallow bowl, whisk together the baking soda and warm water, and then fully immerse the pretzel into the water mixture. Lift and allow excess water to drip off. Place pretzels on the prepared baking sheet, reshape as needed, and sprinkle to taste with coarse salt. Repeat this process with the remaining dough.
Bake the pretzels in the preheated oven for 7 to 11 minutes until golden brown. Once you remove them from the oven, brush the top and bottom of the pretzels with melted butter. Serve warm and enjoy.