Combine the rye flour, milk, onion, honey, unsalted butter and yeast in the bowl of a stand mixer using the dough hook. Allow to rest, 5 to 10 minutes. The mixture may not bubble much but it should start to swell slightly.
Add 2 1/2 cups of the all-purpose flour, the water and caraway seeds. Mix slowly at first and then on medium speed. Add 1/2 cup more of the all-purpose flour and the salt. The dough should come together and have a tacky but not too sticky consistency. Add more of the all-purpose flour, as needed, to get the dough where it just pulls away from the sides of the bowl. Continue mixing until the dough looks smooth, about 5 minutes.
Preheat the oven to 350 degrees F. Turn the oven off 3 minutes after reaching the temperature. Spray the inside of a bowl with nonstick spray. Place the dough in the bowl and cover tightly with plastic wrap. Place the bowl in the turned-off oven. Allow the dough to rise until doubled in bulk, about 1 hour.
After the dough rises, remove the bowl from the oven. Preheat the oven to 350 degrees F.
Punch the dough and transfer to a work space. Separate the dough into 13 pieces, about 3 ounces each, using a bench knife. Use both your hands to roll the dough into tight, uniform balls. Make sure there are no air pockets in the middle of your dough balls.
Roll each ball into 7-inch cylinders using both hands and rolling from the middle out. Make sure the cylinders are even in length or your buns will be wonky. Keep whatever dough you are not immediately working with covered loosely with plastic.
Line a 15-inch-long, or longer, sheet pan with parchment paper or cover well in nonstick spray. Place each dough cylinder side-by-side up the center of the sheet pan, leaving 1/2-inch space between each one. Cover the pan with plastic and allow to rise until the cylinders are touching, about 15 minutes.
Remove the plastic wrap. Combine 1 tablespoon water with the egg white and brush this mixture in an even coat all over the dough. Allow the dough to rise, about 15 minutes.
Bake the dough until the buns are nicely golden brown on top and 190 degrees F in the center, rotating the pan halfway through the baking time, about 20 minutes total. Remove pan from oven and allow the buns to rest in the pan, 10 to 15 minutes.
Transfer the buns in one piece (that’s very important) to a cooling rack to prevent soggy-bottom buns and cool, about 30 minutes.
Transfer the buns to a cutting surface. Slice down the center of each tube of dough using a serrated knife (this is the cool part). You will lose half a tube on each end, hence yielding only 12 buns and 2 awesome slices of rye bread that you will eat immediately. The place where the tubes meet is now the center of the bun. They should split naturally but you can use a serrated knife to help them, if you choose. Note:Alternatively, I’ve found it works very well if you place tubes in a pan with higher sides and Slice between the natural separation. As long as the buns were touching each other while baking, you should still get the “bread wall effect” and the buns will be more even.
Now you have a beautiful bread wall that you can butter and toast. Wrap the buns in plastic or a re-sealable plastic bag until ready to use. Freeze the buns immediately if you will not be using them the same day. Yield: 12 buns and 2 slices.