Peel and cut the potato into 3/4-inch pieces, place in a small pan with the 3/4 teaspoon of salt. Cover with cold water and bring to a boil over high heat. Turn the heat to simmer and cook uncovered for 15 to 20 minutes, or until the potatoes are soft. Drain, cover, and put back on the turned off burner to allow the potatoes to dry off for 5 minutes.
Mash the potatoes and measure out a 1/2 cup of firmly packed potatoes. If you love weighing things, this will be about 6 ounces. Place in your stand mixer fitted with the paddle attachment. Add in your oil, sugar, and salt and on low-speed mix until smooth .
Add in the egg, milk, flour, and yeast in this order. Switch to the dough hook and on low-speed until the dough is coming together.
Knead on medium speed for 10 minutes until the dough is stretchy, smooth, and it is cleaning off the sides of the bowl. The dough will be soft and some may stick to the bottom of the bowl, but that is okay.
Shape the dough into a ball and pour a teaspoon of oil over top. Turn to dough over so it and the sides of the bowl have a light coating of oil. Cover and place in a warm place for an hour or until it has doubled in size. If your kitchen is cold, put the dough in your unheated oven with the light on.
Turn the dough onto a lightly greased surface and gently press out the air bubbles. Using a scale or working by eye, divide the dough into 12 or 24 balls of dough. If using a scale weigh your dough and figure out grams. (My last test batch weighed 960 grams so each of the small rolls weighed 40 grams.)
Shape the dough into smooth balls by working in circles bringing the edges into the middle, either resting the dough in your hand or on the counter. When you have a pretty decent ball, cup your hand over the ball keeping the edges of your hand on the counter, and move in a small circular motion. This will create a smooth surface. If you would like to see a very short demo on shaping, head on over to King Arthur Flour.
Place the rolls side by side in a greased 9" x 13" pan. Cover loosely and allow them to rise for another 30 to 40 minutes until doubled.
Baking
Preheat the oven to 350°F (180°C). If using a butter glaze, melt the butter and gently brush the rolls so as not to deflate them.
Bake for 20 to 25 minutes until golden brown and they reach 195 to 200°F (100°C). If you find they are browning before being done inside, loosely cover them with foil.
Cool the rolls in the pan for 15 minutes, then gently remove them onto a wire rack. When cooled completely serve or wrap them tightly and keep at room temperature for two days. Otherwise, pop them in the freezer. They are best eaten warm/toasted.
Notes
Texture: To keep these rolls light and fluffy, use a bit of oil when shaping the dough rather than flour.
Make in two steps: If you want to break this potato roll recipe in half, pop the dough into the fridge overnight for the first rise. Bring it to room temperature and it if hasn't doubled in size allow it to rest, then continue with the recipe.
Keeping: These rolls freeze beautifully. Break them apart and store them in a freezer bag, squeezing out as much air as possible. They will keep well for two months.
Dairy-free, Vegetarian, and Vegan: Instead of a butter glaze, any non-dairy milk works, but soy milk is ideal. Recipe: 2 tablespoons of non-dairy milk mixed with 1 tablespoon of maple syrup.
Egg-free: Use 1/4 cup full-fat plain yogurt instead of the egg