In a medium bowl, whisk together warm water, sugar, and yeast. Let sit until foamy and bubbly, about 5 to 10 minutes.

Stir in melted butter, 2 1/2 cups flour, and salt. Mix until a shaggy dough forms.

Turn onto a lightly floured surface and knead for 5 to 8 minutes, adding up to 1/2 cup more flour as needed.

Place dough in a lightly oiled bowl, cover with a clean towel or plastic wrap, and let rise in a warm spot until doubled in size, about 30 to 45 minutes.

Punch the dough down and divide into 8 equal pieces. Roll each piece into a long rope about 12 inches long, then cut into 1-inch pieces.

In a wide shallow pot, bring 4 cups of water and baking soda to a simmer.

Working in batches, drop pretzel bites into the simmering baking soda bath for about 15 to 30 seconds.

Remove with a slotted spoon and drain briefly on a wire rack.

Place bites on a parchment-lined tray. Brush each lightly with egg wash and sprinkle with coarse salt.

Preheat the air fryer to 375°F / 190°C for 3 minutes. Arrange pretzel bites in a single layer; cook in batches if needed.

Air fry for 6 to 8 minutes, turning once halfway through, until golden brown and crisp on the outside.

Brush with melted butter and serve warm.