Mix the yeast and honey with the water and let yeast soften for 5 minutes. Mix in the butter, egg, egg yolk, lemon juice, crushed caraway seeds, all of the rye flour, and 2 cups of the wheat flour until smooth, 1 to 2 minutes. Drape with a towel and let dough rest for 20 minutes so the flour can properly hydrate before adding the remaining flour.
Add the salt and remaining flour 1/2 cup at a time, mixing 1 minutes after each add, to form a soft, shaggy dough (up to 5 1/2 cups, in total). (If using a stand mixer, a paddle attachment can work better than a dough hook.) Once the dough holds it's shape (though sticky) and is stable on the mixing paddle, spread 1/4 cup of flour out on the counter. With floured hands, knead dough by folding over and turning 4 or 5 times to form a soft ball.
Note: Because the dough can be tacky, use a pastry knife to help the dough release from the counter rather than adding more flour, which will make the dough heavy. Switching from flour to oil on your hands can help too.
Leave the dough to rest on the counter and cover with a bowl for 5 minutes. With oiled hands, fold the dough 4 or 5 times again, and cover with the bowl to rest again for 5 minutes. Repeat one more time so the dough has 3 folding processes and 3 rests of 5 minutes each.
While you shouldn't need to, at this point, if the dough is still is not holding its shape and is too wet to handle, add 2-3 tablespoons more flour and complete the folding process again with an additional 5 minute rest. Otherwise, after the three folding/resting cycles, place the dough in an oiled bowl, turning over so the oiled surface it face-up.
Cover the bowl with plastic wrap and then a towel and let it rise in a warm spot (above 70°) for 1 to 1 1/2 hours, or until nearly doubled in bulk. (The oven, with the oven light ON is a good place to let the dough rise if the room is cool.)
With oiled hands, punch down dough and divide it into 16 pieces on a lightly oiled surface (using a kitchen scale to measure even bun dough weight is ideal). Shape each piece into a round, smooth ball, place on a parchment-lined baking sheet, and flatten with the palm of the hand to about 3" across (it helps to have oil on your hands when flattening the buns).
It is best if the sides of the buns touch when flattened. Cover with a lightweight smooth towel, tent aluminum foil, or inverted rimmed baking sheet if the room is drafty. Let rise for 30-40 minutes, or almost doubled in size. (Because the dough can be tacky, avoid covering with terry cloth or microfiber cloth towels, which can "grab" the top of the dough.)
Just before placing in the oven, brush the buns gently with the beaten egg white on all exposed sides, sprinkle with caraway seeds. Alternatively, brush with melted butter once the are baked and removed from the oven for soft, matte finish.
Bake the buns in at 375°F until golden in color, about 18 minutes, or until the interior registers 195°F using an instant-read thermometer. Cool the buns on the pan on a wire rack for 10 minutes, then remove from pan and cool completely on the wire rack.
Crush the caraway seeds to release more of the flavor in the seed. Use a mortar and pestle or a heavy cast iron skillet to crack or crush the seeds.
The egg wash helps the seeds to stick and give the buns a shiny crust. Brush with melted butter once removed from the oven for a soft, matte finish.
The weather and type of flour can have an impact on the dough and how tacky it feels. Avoid adding a lot of extra flour to the dough to make it more manageable. Instead, use a pastry knife to scrape and lift it when kneading or switch to an oiled surface and oil for your hands to manage the dough.
To use a kitchen scale when dividing the dough, simply weigh the mass of dough and divide by the number of desired buns. Example: 44.97 ounces divided by 16 (buns) = 2.81 ounces of dough per bun.
To make a dark rye, substitute molasses for the honey.