In a large bowl, combine the warm water and sugar. Sprinkle the yeast on top and let it sit for about 5 minutes until it becomes foamy. Add the melted butter, salt, and flour. Mix until a dough forms and pull away from the sides of the bowl.
Knead the dough on a floured surface for about 5 minutes until smooth and elastic. Place the dough in a greased bowl, cover with a clean towel, and let it rise in a warm spot for about 30 to 45 minutes, or until it has doubled in size.
Preheat your oven to 450°F (230°C). Punch the dough down and turn it onto a flat surface. Divide the dough into 8 equal pieces. Roll each piece into a long rope and use a knife to cut the rope into 1-inch bite-sized pieces.
Bring 10 cups of water and the baking soda to a rolling boil in a large pot. Drop the pretzel bites into the water in small batches for about 30 seconds each. Remove them with a slotted spoon and place them on a parchment-lined baking sheet.
Brush the top of each pretzel bite with the beaten egg wash and sprinkle generously with coarse salt. Bake for 12 to 15 minutes or until the pretzels turn a deep golden brown.
In a small saucepan over medium heat, melt the butter. Whisk in the flour and cook for 1 minute. Slowly pour in the milk while whisking constantly until thickened. Remove from heat and stir in the shredded cheddar cheese until smooth and melted.