For the pretzels:
For the Sauce:
Make the pretzels: Warm the milk in a saucepan until it’s about 110 degrees; pour into a medium bowl and sprinkle in the yeast. Let the yeast soften, about 2 minutes; stir in the brown sugar and 1 cup flour with a wooden spoon. Dice 2 tablespoons butter and soften; stir into the mix. Add the remaining 1 1/4 cups flour and the fine salt to make a sticky dough. Turn the dough out onto a lightly floured surface and knead, adding more flour if needed, until smooth but still slightly tacky, about 5 minutes. Shape into a ball, place in a lightly greased bowl and cover with plastic wrap. Let rise in a warm spot until doubled in size, about 1 hour.
Preheat the oven to 450 degrees and grease a large baking sheet. Punch the dough to deflate it, then turn out onto a lightly floured surface. (If the dough seems tight, cover and let rest until it relaxes.) Divide the dough into 6 pieces. Roll and stretch each piece with the palms of your hands into a 30-inch rope, holding the ends and slapping the middle of the rope on the counter as you stretch. Form each rope into a pretzel shape.
Dissolve the baking soda in 3 cups warm water in a shallow baking dish. Gently dip each pretzel in the soda solution, then arrange on the prepared baking sheet and sprinkle with the coarse salt. Bake until golden, 10 to 12 minutes.
Prepare the sauce: Combine the mayonnaise, mustard, brown sugar and vinegar in a bowl. Cover and refrigerate.
Melt the remaining 8 tablespoons butter in a shallow dish. Dip the hot pretzels in the butter, turning to coat; place on a wire rack to let excess butter drip off. Serve the pretzels warm with the sweet mustard sauce.