Combine the dry ingredients. Add the rye flour, wheat germ, caraway seeds, yeast, salt, and 1 cup of the all-purpose flour to the bowl of a stand mixer fitted with a paddle attachment. Mix to combine and set aside.
Warm the wet ingredients. Add the milk, water, butter, and honey to a small saucepan. Heat over low heat just until the mixture reaches 120-130°F.
Mix the dough. Pour the wet mixture into the bowl with the dry ingredients. Beat on medium speed for 3 minutes.
Add the remaining flour. Add just enough of the remaining all-purpose flour to form a soft dough. It will be sticky — do not add too much flour or the rolls will be dense.
Knead the dough. Switch the paddle attachment for a dough hook and knead on low to start. After a couple of minutes, increase the speed to medium and knead for 6-8 minutes.
First rise. Cover the bowl and allow the dough to rise in a warm place until doubled in size.
Divide and shape. Turn the dough out on a lightly floured surface and divide into 18 pieces, about 2 ounces each. Roll into round balls — cup your hand over each ball and swirl it on the countertop to create evenly shaped rolls.
Arrange on pans. Place the rolls on 2 large, generously greased baking sheets, 9 per pan with about 2 inches between dough balls. Cover with a clean dry kitchen towel and let rise in a warm place until nearly doubled in size.
Preheat the oven. Preheat the oven to 400°F. Position two oven racks: one in the second space from the top and the other two spaces above the bottom.
Apply the egg wash. Whisk the egg and water in a small bowl. Brush over the tops of the rolls, avoiding pooling at the bottom, which could make them stick to the pan. Sprinkle with additional caraway seeds and flaky sea salt if desired.
Bake the rolls. Bake until lightly browned, rotating the pans halfway through and switching racks. Serve hot with lots of butter.