To begin, combine the rye flour, oil, molasses, salt, yeast, caraway seeds, water, and 2 cups of the bread flour in the bowl of your stand mixer.
Turn on your mixer (fitted with the dough hook) and while it mixes, gradually add in the remaining 2 cups of bread flour. Let the mixer run for 6 minutes.
Take the dough out, form it into a ball, and then place it in a lightly oiled bowl. Turn the dough to coat the top with oil. Cover the bowl with a tea towel and place it in a warm location to rise for about an hour, or until the dough has dobled in size. (See the note in the post about rising the dough.)
Turn the dough out onto a lightly floured surface and divide the dough in half.
Pat one half of the dough out into a rectangle that is about 7×14 inches. Starting on a long side, roll the dough tightly into a log. Place the loaves on a baking stone or baking sheet tucking the ends underneath. The baking sheet/stone should be lined with parchment paper or lightly greased. Repeat with the other dough portion.
Cover with the tea towel again and place it back in its warm location to rise again.
After about 30 minutes, if you touch the dough a light indentation should remain. Set your sheet out on the counter and begin heating your oven to 400°(F).
Use a very sharp knife, bread lame or razor blade cut three or 4 slashes on the top of the loaves at an angle.
In a small dish, whisk an egg white until frothy. Brush the egg white over the loaves.
When your oven is fully heated, bake the bread for 30-35 minutes or until they sound hollow when tapped.