In a large bowl, stir together the warm water, sugar, and yeast, then let it sit for about five minutes until it turns foamy on top. This tells you the yeast is alive and ready to work. If it doesn't foam up, your water was likely too hot or too cold.
Add the flour, salt, and melted butter into the yeast mixture, and stir until a shaggy dough forms. Turn it out onto a floured surface and knead for about five minutes until it feels smooth and slightly elastic. This step builds the structure that gives pretzels their chew.
Place the dough in a lightly oiled bowl, cover it with a towel, and let it sit somewhere warm for about an hour. It should roughly double in size during this time. Don't rush this part, since a properly risen dough makes a huge difference in the final texture.
Punch down the risen dough and divide it into eight equal pieces. Roll each piece into a long rope, then twist it into a classic pretzel shape, tucking the ends underneath to seal them.
Warm the water and baking soda together in a saucepan until just simmering, then dip each shaped pretzel into the solution for about 20 seconds. This quick bath is what gives pretzels their signature dark, chewy crust once cooked.
Brush each pretzel with the beaten egg, sprinkle with coarse salt, and place them into your air fryer basket in a single layer. Cook at 375°F for 7 to 9 minutes, checking around the halfway point, until they're deep golden brown all over.