Put the bread flour, rye flour, yeast, salt, and caraway into the bowl of a stand mixer. Use the dough hook attachment to stir together.
At low speed, add the warm water.
When the last piece of butter has been added, increase the mixer speed to medium and knead the dough for 7 minutes. The dough should be elastic, but very tight and firm. If the dough seems too sticky, add more flour 1 tablespoon at a time, until the dough pulls away from the sides of the bowl.
Transfer the dough to a clean work surface and cover it with plastic wrap to keep it from drying out while you portion pieces.
Divide the dough into walnut-sized pieces (20g each if you have a scale), and arrange them on a couple of baking sheets; don’t let the pieces touch.
Cover the pans with plastic wrap and set them aside in a warm place to proof (rise) for about an hour.
When you are ready to start shaping the pretzels, first preheat the oven to 400 degrees F.
Make a blanching solution by bringing the 10 cups of water and the baking soda to a boil in a large pot. (When choosing the pot, make sure the solution will be at least 2 ½ inches deep.)
The solution should be boiling when you start shaping the pretzels.
Meanwhile, generously grease a baking sheet with oil or nonstick spray and set it aside.
Make an egg wash by combining the egg with the 1 tablespoon water; whisk well, until all traces of the egg white have disappeared. Keep the egg wash in the refrigerator until ready to use.
To shape the pretzels, roll each little piece of dough into a 5- to 6-inch snake and tie it in a knot.
Once you have rolled and tied 10 pretzels, immediately put them into the boiling solution. Boil each batch of pretzels for 30 seconds, dunking them underwater with a large spoon as they cook. After the 30 seconds, transfer the pretzels to a wire rack using a slotted spoon.
With the pretzels still on the wire rack, generously brush them with the egg wash, then transfer them to the well-greased baking sheet.
Sprinkle the pretzels with coarse sea salt and bake them for 20 minutes, rotating the pan after 10 minutes. If your oven has a convection fan, turn it on for the last 2 minutes of baking to increase browning.
Cool the pretzels on a wire rack. They are best eaten the day they are made, but you can store any leftovers in an airtight container.