Combine water and butter in a small saucepan; heat over low until butter is melted. Remove from heat and let cool 5 minutes, or until mixture reaches 110-120°F.
In bowl of stand mixer fitted with a dough hook attachment, combine 2 cups flour, yeast, salt, and sugar. Add water mixture and mix until combined to form a wet dough.
With mixer on medium low, add remaining flour 1/4 cup at a time until dough clears the bowl but is still slightly sticky to the touch. You may not need to use all of the flour. Continue to knead until dough is smooth and elastic, about 5-7 minutes.
Place dough in a greased bowl and cover with plastic wrap. Let rise in a warm place until doubled, about 1 hour.
Gently punch down dough and knead a few times. Tear a small handful of dough off with your hand and roll into a long thin rope about 1/2 inch wide. Use a sharp knife to cut into 1-inch pieces. Repeat with remaining dough.
Preheat oven to 450°F; line 2 sheet pans with parchment paper.
Pour 10 cups of water into large pot; stir in baking soda and bring the mixture to a boil over medium-high heat.
Drop dough pieces into water in small batches; cook 30 seconds. Remove with a slotted spoon and place on prepared pans.
Combine egg with 1 tablespoon water and brush egg wash over pretzel bites.
Bake until golden brown, about 10-12 minutes.
Combine sugar and cinnamon in large bowl. Drizzle warm pretzel bites with butter and transfer to bowl. Toss with cinnamon-sugar mixture until well-coated. Pretzel bites are best enjoyed warm, but will keep in an airtight container for a few days.