Combine warm water, yeast, and honey in the bowl of a standing mixer.. Let sit for 10 minutes to proof.
Whisk rye flour, bread flour, salt and caraway seed together in a separate bowl.
Slowly add the flour mixture to the yeast mixture in the bowl of the standing mixer. Use the dough hook attachment to stir everything together.
Add the butter and the egg to the mixture and continue to mix until the dough pulls away and cleans the side of the bowl.
Turn dough out onto a lightly greased surface and shape into a round.
Place round in a lightly greased container. Cover and let rise until double in size (about 2 hours).
Turn dough out onto a lightly greased surface and shape into a sandwich loaf.
Transfer dough to a 9" by 5" bread pan and let rise until it reaches the lip of the pan, about 1 1/2 hours.
In a small bowl, mix together the egg white with about 1 tablespoon of water to create an egg wash. Brush over the loaf and sprinkle with extra caraway seeds.
Preheat oven to 400° F (204° C) and bake for 20 minutes.
Tent your loaf with aluminum foil and bake for another 25 minutes.
Turn out loaf onto a wire cooling rack and allow it to cool completely.
Slice, serve, and enjoy!
Notes
You don't have to use all of the egg wash on the loaf! Just brush enough on to cover the top.