For the challah buns: In a medium bowl, combine the flour, sugar, yeast and salt. Whisk to combine. In a liquid measuring cup, combine the warm water, oil and eggs. Beat to mix and pour into the dry ingredients. Stir to form a dough. Knead either in the stand mixer or by hand on a floured surface, adding flour as needed to prevent sticking, until smooth and just slightly sticky, 7 to 10 minutes. Grease the inside of a large clean bowl and transfer the dough to the bowl. Cover with a towel and let the dough rise until doubled in size, 2 to 3 hours.
For the baking soda bath: While the dough is rising, bake the baking soda. Preheat the oven to 250 degrees F.
Spread the baking soda out on a rimmed baking sheet and bake for 1 hour. Let it cool and set it aside (even though the baked baking soda isn't as strong as lye, it could potentially irritate your skin, so avoid touching it once it's out of the oven).
Line a baking sheet with parchment paper.
Split the dough into 8 equal portions and keep it covered when you're not working with it. Working with one piece of dough at a time, stretch it into a ball by tucking the top surface under and pinching to seal, then place it on a baking sheet. Continue with the remaining dough pieces, placing them 1 1/2 inches apart on the baking sheet. Cover and let rise for 20 more minutes.
Increase the oven temperature to 375 degrees F.
Place the baked baking soda in a large bowl or casserole dish and add 2 cups (472 grams) of water. Stir it to dissolve the baking soda. (I've never been able to get all of it to dissolve, so a few baking soda chunks are totally ok as long as they don't stick to the dough.)
For the topping: Using gloved hands or tongs and working in batches, immerse the rolls in the baking soda bath for about 2 minutes on both sides. Pat them dry with a paper towel and place them back on the baking sheet. Brush with the egg wash and sprinkle with coarse pretzel salt. Cut 2 shallow slits in an X shape on top and bake until golden brown; begin checking for doneness at 16 minutes.