In a medium saucepan, heat buttermilk and water together until warm (105 degree F to 115 degree F). In a large mixing bowl, pour buttermilk mixture over lard or shortening, yeast, sugar, caraway seed, and salt. Stir in 2 cups of whole-wheat flour and all the rye flour. Stir in as much remaining whole-wheat flour as you can.
Turn dough out onto a surface that's dusted with whole-wheat flour. Knead in enough of the remaining whole-wheat flour to make a moderately stiff dough that's smooth and elastic (6 to 8 minutes total). Dough may be slightly sticky because of the rye flour.
Shape into a ball. Place in a very large greased bowl, turning once to grease surface. Cover; let rise in a warm place until doubled (50 to 60 minutes).
Punch dough down. Turn out onto a lightly floured surface. Divide in half. Cover and let rest for 10 minutes.
Form dough into two round loaves. Place on a greased baking sheet. Flatten into 5-inch circles. Cover and let rise in a warm place until nearly doubled (about 30 minutes). Score tops.
Bake in a 350 degree F oven for 35 to 40 minutes or until done. If you like, for glaze, combine egg white and 1 tablespoon water; brush over loaves during the last 5 minutes of baking. (If you don't brush with egg-white mixture, brush with melted butter after baking, if you like.) Remove from baking sheet immediately. Cool.