In a large bowl, combine the warm water, active dry yeast, and brown sugar. Let it stand for 5 minutes until foamy and bubbly. Add the melted butter, all-purpose flour, and salt. Stir until a shaggy dough forms, then knead on a lightly floured surface for about 5 minutes until the dough is smooth and elastic. Cover with a towel and let it rest for 10 minutes.
Divide the dough into 8 equal pieces. Roll each piece into a long rope about 1/2-inch thick, then use a knife or bench scraper to cut each rope into 1-inch pieces.
Bring 4 cups of water to a boil in a large pot and stir in the baking soda. Working in batches, drop the dough pieces into the boiling baking soda bath for 30 seconds, then remove with a slotted spoon and place on a parchment-lined tray.
This quick bath is what gives the pretzel bites their signature chewy crust and deep golden-brown color.
Preheat the air fryer to 375F. Arrange the boiled pretzel bites in the basket in a single layer with a little space between each one. Work in batches to avoid overcrowding.
Air fry for 7 to 8 minutes, shaking the basket gently halfway through, until the pretzel bites are deeply golden brown all over.
Immediately brush the hot pretzel bites with melted butter and sprinkle generously with coarse sea salt while the butter is still glistening.
Serve warm with cheese sauce or whole grain mustard for dipping.