Combine the flour, yeast, sugar, salt, 1 cup warm water, and 3 tablespoons melted butter in the bowl of a stand mixer with the dough hook attached. Mix on medium to medium-high speed until a dough forms. This should take about 4 or 5 minutes. The dough will be soft and smooth.
Cover the dough with plastic wrap or a towel, and allow to rest for 1 hour. The dough won't double in size but will be slightly puffy.
Preheat oven to 475°F (245°C). Line a half sheet pan with a silicone liner or parchment paper rated for high temperatures.
Bring 8 cups of water to a boil in a large pot. Very slowly add the baking soda.
Transfer the dough to a lightly floured surface. Divide into 8 equal portions, and roll each portion into a rope about 12 inches long. Cut each rope into 1-inch pieces. (The dough is quite elastic, so you may not get 12 pieces from each portion.)
Add the pieces of dough one at a time to the baking soda water. Do this in a few batches so the pan doesn't become too crowded. Leave the dough in the water for 30 seconds.
Remove the dough from the water using a slotted spoon, tongs, or a spider strainer. Place on the prepared pan, making sure the pieces aren't touching. (Use a second pan if needed.) Brush with the beaten egg.
Bake for 10 to 12 minutes, or until the pretzel bites are golden brown.
Remove the pan from the oven. Brush the pretzels with melted butter and sprinkle coarse salt over the top.