Prepare the Dough:

1. In a large mixing bowl, combine the warm water and light brown sugar, stirring until the sugar dissolves.
2. Sprinkle the active dry yeast over the water mixture and let it sit for 5 minutes until it becomes foamy.
3. Add the melted butter and kosher salt to the yeast mixture, stirring to combine.
4. Gradually add 4 ½ cups of all-purpose flour, mixing until the dough comes together.
5. If the dough feels excessively sticky, gradually incorporate additional flour.
6. Turn the dough out onto a lightly floured surface and knead for 5-7 minutes until the dough is smooth and elastic.
7. Lightly grease a large bowl with vegetable oil, place the dough in the bowl, and cover it with plastic wrap or a kitchen towel.
8. Let the dough rise in a warm spot for about 1 hour, or until it has doubled in size.

Shape and Boil the Pretzel Bites:

1. Preheat your oven to 425°F (220°C) and line a baking sheet with parchment paper.
2. Bring 3 quarts of water to a boil in a large pot.
3. Once boiling, slowly add the baking soda (be careful as it may foam up).
4. Punch down the dough and divide it into 6 equal pieces.
5. Roll each piece into a rope, about 12-14 inches long, and cut each rope into 1-inch bite-sized pieces.
6. Working in batches, carefully drop the pretzel bites into the boiling water for about 30 seconds.
7. Use a slotted spoon to remove the bites and place them onto the prepared baking sheet.

Bake the Pretzel Bites:

1. Brush the tops of the boiled pretzel bites with the beaten egg wash, then sprinkle with coarse sea salt.
2. Bake the pretzel bites in the preheated oven for 12-15 minutes, or until golden brown and puffed.

Make the Cheese Sauce:

1. In a small saucepan, melt the butter over medium heat.
2. Add the flour and cook for 1 minute, stirring to form a roux.
3. Gradually whisk in the milk, continuing to stir until the mixture thickens, about 2-3 minutes.
4. Remove the saucepan from the heat and stir in the grated cheddar cheese until smooth and melted.
5. If the sauce is too thick, you can add a little more milk to achieve your desired consistency.

Serve:

1. Serve the soft pretzel bites warm, with the creamy cheddar cheese sauce on the side for dipping.