In a small saucepan, place in the potato and water. Cover and bring to a boil, then turn off the heat and let the potato steep in the water for about 5 minutes, remove and discard the potato. Cool the water until lukewarm, between 105°-110°. (You will use it at this temp. when you add the yeast.)
In a separate saucepan, or in the microwave, heat the butter and buttermilk together until the butter melts. Cool until it’s lukewarm.
In a large bowl, place in the Active Dry Yeast and Sugar, and then pour in the potato water, (at 105°-110°). Stir to combine and let rest for about 5 minutes to let the yeast activate.
To the mixture add the Butter and Buttermilk, and then the salt. Stir everything to combine.
To the mixture, add the Rye flour. Mix this until it is very smooth.
To this mixture add half of the All-Purpose Flour. Continue to mix and combine. You will probably need to use your hands at this point. From here, gradually add in the rest of the All-Purpose Flour a little at a time until you get a medium-stiff dough. (You may not use it all, or you may need to add a little more. It depends how it comes together in your environment.) Turn the dough out on a floured board and kneed until it is very smooth, adding more flour to the board if the dough seems to be too sticky. (It should be a little tacky, but not be sticking to the surface.) This should take about 10 minutes. Use your hands to rub olive oil all over the inside of a large bowl. (Select one twice the size of the dough to give it room to rise.) Place the dough in the bowl, cover with plastic wrap and a kitchen towel, and let rise until doubled in size. Approximately 90 minutes.
After the dough rises, punch the dough down. (Use your fist to punch down the dough in the middle.) Cover the bowl again with the plastic wrap and kitchen towel and let it rise again, until double.
Oil or spray two pie plates. Divide the dough into two pieces and form them into round loaves, and place each in a pie plate *You can also make them into a loaf shape and place them each into a prepared loaf pan.
Cover the loaves with a kitchen towel and allow to rise for 30-40 minutes.
Preheat your oven to 375°F
Brush the loaves with a layer of the beaten egg wash. Place the loaves on the middle rack in the oven and bake for 45 minutes, or until nicely brown on top. Remove them from the oven and brush softened butter over the top.