Peel and dice potato into 1-inch pieces. Place in medium sauce pan and cover with water to about 1 inch over the potatoes. Place it over medium heat and bring to a boil. Boil until potatoes are soft, about 8 minutes. Reserve 1 cup of potato water. Drain potatoes and place back into sauce pan.
Place saucepan over low heat and stir potatoes gently. This process will remove excess water from the potatoes. It should only take 30 seconds. Remove from heat and push potatoes through a food mill or masher. Measue 1 cup of potato, firmly packed into the measuring cup. Toss leftovers. Add butter to 1 cup of potato in a small bowl and toss together until butter is melted and combined with potato. Set aside until cool.
Attach bowl to stand mixer and fit mixer with dough hook (you can mix by hand, if you don't have a mixer). Add flour, yeast and salt and mix on low to combine. Add cooled potato mixture and mix together. In a separate bowl, mix 5 tablespoons of reserved potato water, sugar, and 1 large egg. Slowly add to flour mixture while mixer is on low. Increase speed to medium low to help everything combine.
Once dough has formed and no longer shaggy, turn up speed to medium and knead dough for 8 minutes. The dough is ready when it's soft and slightly elastic. Place dough ball in a greased bowl and cover with a damp towel. Let it rise for 1 hour. It should be double in size.
Punch down risen dough and remove from bowl. Place on clean counter with a little sprinkled flour. Cut dough into 3 even slices. Cut each slice into 4 even pieces. You should have 12 even pieces. Roll each piece into a ball, tucking under the edges to form a ball. Place all the ball into a greased 13 x 9 pan. Cover with a towel and let them rise again for 45 minutes. At the 30-minute mark, preheat oven to 425 degrees.
Remove towel from rolls and brush with egg wash. Sprinkle flakey sea salk on top of each roll. Bake in preheated oven for 15-20 minutes, or until deep brown on top. Remove from oven and let cool for 10 minutes. Serve warm with butter. Or use as slider buns for mini sandwiches!
Leftover rolls can be stored in a plastic storage bag for a couple days or frozen for a month or so. Rolls can be reheated in microwave for 30 seconds.