Place cubed potato in a medium stock pot, cover with 1 inch of cold water. Bring to a boil over high heat, reduce to simmer, then cook until tender -- should be able to pierce with a paring knife or fork. 8 to 10 minutes.
Pour 177 grams (6 ounces) of the potato water into a measuring cup and let cool completely. Reserve the rest of the water as you may need more.
Drain the rest of the water from the pan and let the potato chunks cook over low heat, shaking until all surface moisture is gone. 30-60 seconds.
Run potatoes through rice mill or mash well with a potato masher.
Measure out 1 cup of the potatoes and put into a separate bowl.
Stir the butter into the 1 cup of potatoes until melted and let the whole thing cool completely. Resist the urge to eat the mashed potatoes.
Keep the remaining mashed potatoes for whatever evil purpose you have.
In the bowl of a stand mixer, whisk together the flour, salt, and yeast.
Add the mashed potatoes and mix well.
Using the dough hook on low speed, slowly add the potato water and mix until a dough forms and no dry flour remains, scraping down the sides as needed. ~2 minutes. Depending on humidity, you may need more than the 177ml kept above.
Increase speed to medium and knead for another 8-10 minutes until dough is smooth, elastic, and cleaning the sides of the bowl.
Reduce the speed to low and add the dill. Mix until incorporated.
On a lightly floured counter knead to make a smooth, round ball. Transfer dough to an oiled bowl, seam side down, and cover with plastic wrap. Let rise until doubled: 60-90 minutes.
Grease a 1 pound (8½" x 4½") loaf pan.
Gently deflate dough and turn out onto a lightly floured surface.
Press and stretch dough until you've got about an 8x6" rectangle.
Roll dough away from you to form a cylinder. Keep taut. Pinch seam and place loaf seam side down into loaf pan.
Cover loosely with greased plastic wrap and let rise until loaf is ~½" above the edge. Dough should spring back slowly when pressed with your knuckle. This should take 20-30 minutes.
Adjust oven rack to middle position and heat oven to 350℉ (~175℃)
Mix egg, water, and salt in a small bowl. Brush top of loaf with egg mixture.
Using a sharp knife or lame, make a ½"-deep slice end to end.
Bake until golden brown, 45-50 minutes, and thermometer registers 205-210℉ (96-100℃) in the middle of the loaf.
Let loaf cool in pan for ~15 minutes before removing to a wire rack to cool completely (2-3 hours)