Pour the lukewarm water into a bowl and sprinkle in the yeast. Whisk it for about one minute so it dissolves. This step makes sure your dough rises well.
Add the melted butter, brown sugar, and salt to the yeast mixture. Stir until everything is blended. The mixture should look slightly foamy and smell yeasty.
Stir in three cups of flour, one cup at a time. Then add another 3/4 cup. If the dough is still sticky, add flour 1/4 cup at a time until it springs back when poked.
Turn the dough onto a lightly floured surface and knead for two minutes. Shape it into a ball and place it in a lightly greased bowl. Cover it and let it rest for 10 minutes.
While the dough rests, bring a large pot of water to a boil and stir in the baking soda. Preheat your oven to 425°F. Line two baking sheets with parchment paper.
Divide the dough into six equal pieces. Roll each piece into a long rope. Cut the ropes into 1 1/2-inch bites.
Drop about 10 bites into the boiling water at a time. Let them boil for 20 seconds, then lift them out with a slotted spoon. Place them on the baking sheets with space between each one.
Brush the tops with the beaten egg and sprinkle with coarse salt. Bake for 12 to 15 minutes until deep golden brown. Rotate the pans halfway through for even color.
While the bites bake, heat the milk in a small saucepan over medium heat. Wait until small bubbles form on top. Do not let it boil hard.
In a second saucepan, melt the butter over medium heat. Whisk in the flour and cook for two to three minutes until lightly browned. This builds the base for a smooth, thick sauce.
Slowly pour the warm milk into the roux while whisking nonstop. Cook for about five minutes until thickened. Stir in the cheddar until smooth, then season with salt and pepper.
Pile the warm pretzel bites in a bowl and pour the cheese dip into a small dish. Serve while everything is hot. The bites are softest fresh from the oven.