In a large mixing bowl, combine the warm water and sugar. Sprinkle the yeast over the surface and let it sit for 5 minutes until it becomes frothy.
Add the flour, salt, and vegetable oil to the yeast mixture. Stir until a smooth dough forms.
Turn the dough onto a lightly floured surface and knead for 5 minutes until it is elastic. Place the dough in a greased bowl, cover it with a clean towel, and allow it to rise in a warm spot for 1 hour or until it has doubled in size.
Preheat the oven to 450°F (230°C) and line a baking sheet with parchment paper.
In a large pot, bring the water and baking soda to a boil.
Divide the dough into 8 equal portions. Roll each portion into a rope, then cut into 1-inch (2.5 cm) bite-sized pieces.
Carefully drop the pretzel bites into the boiling water and cook for 30 seconds. Remove them with a slotted spoon and transfer them to the prepared baking sheet.
Brush each pretzel bite with the beaten egg and sprinkle with coarse salt.
Bake for 10-12 minutes until golden brown.
While the pretzel bites are baking, prepare the cheese sauce. In a small saucepan, melt the butter over medium heat. Stir in the flour and cook for 1 minute until it turns golden brown.
Gradually whisk in the milk and continue cooking until the sauce thickens. Stir in the shredded cheddar cheese until melted and smooth. Season with salt and pepper to taste.
Serve the pretzel bites warm with the cheese sauce for dipping.