Place warm water in a mixer bowl and sprinkle some yeast over it. Let sit for 10 minutes until foamy. This step is for “proofing” the yeast; if it doesn’t dissolve and become slightly foamy, then the yeast is dead. In this case, repeat with new yeast.
Add sugar, salt, melted butter, and 2 ¾ cups of flour to the water-yeast mixture and stir until combined. Fit your mixer with the dough hook attachment and mix on a low speed until the dough comes together and pulls away from the sides of the bowl. If you don’t have a mixer, mix by hand or use a wooden spoon. If the dough is still wet at this point, add more flour – 2 tablespoons at a time, up to 3 cups total. Keep kneading on low speed in the mixer for 5 minutes, or by hand on a floured surface for 7-8 minutes. The dough should be soft, smooth, and just slightly sticky.
Shape the dough into a ball. Lift it and spray the bowl with a non-stick cooking spray to prevent the dough from drying out. Turn the dough over to coat both sides. Cover the bowl with plastic wrap and let rest in a warm place* for about 1 hour, or until the dough doubles in size.
When it’s finished rising, gently punch to remove air. Divide into 8 equal pieces.
Roll each piece into a rope (1cm width, 50cm long) against the counter using the palms of your hands. Lift the ends of the rope toward the top of your work surface to form a U shape, then cross them. Cross one more time to make a twist, then bring them back down over the bottom loop (towards yourself) to form a traditional pretzel shape (see the post above for a picture demonstration). Place on a baking sheet lined with parchment paper and continue shaping the rest. When all pretzels are shaped, cover them loosely and set them aside to rise until puffy, about 20 minutes.
Preheat the oven to 450F.
Next, prepare the water bath. Place 6 cups of water and baking soda in a large, wide pot. Bring to a boil. Reduce the heat to medium to maintain a simmer. Using a large slotted spatula, and working with 1-3 pretzels at a time (depending on how large your pot is and how many pretzels will fit), place pretzels into the water and let simmer for 30 seconds. Flip pretzels over and let simmer for another 30 seconds. Lift the pretzels and allow any excess water to drip off. Return pretzels to baking sheet.
Brush pretzel tops with egg and sprinkle with coarse salt. Bake until deep brown and glossy – 12 to 15 minutes. Transfer to a cooling rack. Serve warm or at room temperature.
Pretzels are best eaten the same day they are made, but can be frozen for up to 2 months in an airtight container.