For the homemade pretzels:

1. Stir the water and sugar in the bowl of a stand mixer and sprinkle the yeast on top (if not using active dry, see Notes below). Let sit until the mixture begins to foam and turn creamy, about 5 minutes.
2. Add 4 cups of flour, salt, and melted butter and mix on low using the dough hook attachment. If the dough seems extremely sticky, add a little bit of the extra flour until it's slightly tacky but no longer sticky.
3. Increase speed to medium and knead until the dough is smooth and cleans the sides of the bowl. This should take about 5 minutes. It will come together eventually. 
4. Transfer the soft pretzel dough to the counter and form into a loose ball.  Place it in an oiled large bowl and cover it with plastic wrap or a clean kitchen towel. Let the dough rise in a warm place until doubled in size, about 1 hour. The time will differ depending on the temperature of the room. If it shows very little activity after half an hour, cover the bowl with a blanket or take it to a warmer room in your house. 
5. Line a baking tray with parchment paper, lightly brush it with vegetable oil, or use cooking spray. Set aside.
6. When the dough has doubled, turn it out onto a lightly floured work surface.
7. Divide into 8 to 12 portions using a bench scraper. Eyeball the size or use a scale if you want equal pieces that weigh the same. For large, fat pretzels like the ones in the images divide in 8. Divide in 12 pieces for thinner, regular-sized pretzels. 
8. Roll each portion into a rope that is 18" to 24" long. For the classic pretzel shape: lift the ends of each piece of dough to form a sort of U-shape, and then twist the two sides around each other and bring them down to form a pretzel shape. 
9. Place on the baking tray and repeat with all of the dough portions. Let sit uncovered while you prepare the water and preheat the oven.
10. Preheat the oven to 450°F/220°C.
11. Bring 10 cups of water to a boil in a large, wide pot with high sides (like a stew or soup pot). Once it boils, add the baking soda gradually to prevent overflow.
12. When it returns to a boil, lower the pretzels one at a time into the boiling water for 20-30 seconds each (no more!), flipping over halfway. Use a wide, flat-slotted spoon or spatula to lift them from the water. Shake the spatula a little to let as much water as you can fall back into the pot, and place the pretzels back onto the prepared tray.

Egg wash:

1. Once all pretzels are on the tray, beat the egg yolk with the water and brush the top of each one.
2. Sprinkle with coarse or kosher salt (parmesan, poppy or sesame seeds).
3. Slide into the preheated oven and bake until the pretzels turn a deep golden brown, about 15 minutes. You can do a test bake with one and adjust the oven if needed. The inside needs to be fully baked as the outside turns a wonderful golden brown. 
4. Transfer to a cooling rack and allow to cool until warm enough to eat.
5. Store leftovers in an airtight container for 2 days or freeze for a month. Always reheat them before eating. 

For the sauce:

1. Heat beer in a small saucepan.
2. When it starts to bubble, add the rest of the ingredients and whisk until well combined.
3. Let boil for 1 minute and serve with warm baked pretzels.