In the bowl of your stand mixer (with dough hook attached), combine the water, butter, yeast, brown sugar and granulated sugar. Let stand for 5 minutes or until the mixture begins to foam.
With the speed set on low, add the fine sea salt and the flours until all has been added. Increase the speed to medium, and knead for 5-6 minutes. The dough should start to pull away from the sides of the bowl and form a ball. If the dough is too sticky, add more flour until the dough is easy to handle.
Drizzle a tablespoon of canola oil into a large bowl. Remove the dough from the stand mixer and form it into a ball. Place the dough into the large bowl and toss so that it is evenly coated with canola oil. Cover the bowl with plastic wrap or a damp towel and place it in an unheated oven to rise for 1-2 hours, or until it has doubled in size.
Once your dough has risen, remove it from the bowl and transfer it to a work surface. Cut it into 8 equally-sized pieces.
Preheat your oven to 425 degrees F and line 2 large baking sheets with silicone baking mats or parchment paper. Set aside.
In a large pot, bring 9 cups of water to a rolling boil.
While you wait for you water to come to a boil, take each piece and roll out into a very thin rope. Lay the rope horizontally in front of you and cross the right side over the left. Twist the ends once or twice and flip the ends toward you to form a traditional pretzel-like shape. Press the ends into the bottom of the pretzel to seal. Repeat this step for the rest of the dough. (For visuals, see above for a link.)
Once you have formed your pretzels, add the baking soda VERY CAREFULLY and VERY SLOWLY to the boiling water until all of it has been added. One or two at a time, add the pretzels to the water and boil for 30-45 seconds. Remove the pretzels from the water with a slotted spoon and transfer to your prepared baking sheets. Repeat this step for the rest of your pretzels.
Brush your pretzels with the prepared egg wash and sprinkle generously with coarse sea salt. Bake the pretzels at 425 degrees F for 12-15 minutes or until they are brown on top. You may need to switch the two baking sheets halfway through the baking process if your oven does not cook evenly. Remove from oven and let cool for 2-3 minutes before transferring to a wire rack to cool for 10 minutes more.
For the spicy beer cheese sauce (this can be prepared while your pretzels are baking), melt the butter in a large saucepan over medium heat. Slowly whisk in the flour and cook for 1-2 minutes or until a nutty aroma fills the air and your roux is a golden color. Slowly whisk in the beer and continue to let it cook, whisking frequently, for 3-4 minute. Next, add the milk and whisk again to combine. Bring to a boil, then lower the heat to a simmer, and allow for it to cook until the sauce has thickened and is at a consistency in which you desire. Stir in the hot sauce, cayenne pepper, and a dash of salt and pepper. Remove the sauce from the heat, and add the cheese in half cup increments, making sure the cheese has completely melted before you add the rest.
Serve the spicy beer cheese sauce with the pretzels immediately, or store the pretzels in an air tight container where they will keep for about a week.
The cheese can be stored in the fridge for about a week also. When you go to serve it, add it to a small saucepan over low heat to warm gently.