I love baking and these rolls are simple, moist and delicious. If I am busy, I put the ingredients into the bread machine bucket and let it do half the work. But I check it after the first few minutes to see if it needs a tablespoon or more of extra flour. When it is a doughy ball, tacky but not too sticky, it is good to go. Or use your standing mixer or just mix it by hand. Its all good.
If you are using a bread machine, put all the wet ingredients in the bread machine bucket, then the dry. Set for dough only cycle. After the proofing, take out the dough for shaping. (See below)
If you are doing this by hand or mixer, whisk the egg in with the milk. Add the sugar, potato and 1 cup of flour, mixing together. Then add the yeast. Cover and let rest for about 20 minutes to give the yeasty-beasties time to wake up and get busy. If you are using a standing mixer, use the dough hook and mix in the butter, salt and the rest of the flour. It only needs mixing until it is smooth and pulls together. It does not need to be kneaded much to develop the gluten since they are rolls not loaves of bread. That is also why you do not need to use bread flour for rolls. Soft rolls do not need a high protein flour. You can use bread flour, but you don't have to. If you are adding whole wheat flour as part of it, add it early on and let the dough rest. Whole wheat flour takes longer to absorb the liquid. If you do not let it rest 10 minutes or so, you may be in danger of adding too much flour. No matter what the recipe says, you tell if the dough needs more flour by feel. If it is too dry, mist it with water and keep kneading it in. If it is way too sticky, add a tablespoon of flour in at a time until it is tacky and smooth.
Use the mixer or a danish whisk or wooden spoon to mix the wet and dry ingredients together until cohesive. If kneading by hand, put the dough on a floured surface and just knead a few times to make a springy, tacky dough. DO NOT add too much flour. Keep it slightly sticky for moist rolls. Brush with oil and cover with plastic wrap and let rise 45 minutes to an hour. They should puff up and almost double in size. Do not let it rise more than that or you risk weakening the dough.
After the dough is proofed, (risen) gently push it down and divide in half. Then in half again and again until you have about a dozen roll size pieces. Shape each one into a ball with a taut 'skin'. You can do this by putting the little ball of dough on the counter and cup your hand over it, using the palm of your hand and rotating the dough round and round in quick little circles, until it is round and smooth with a little belly button on the bottom. You do not want to flour this counter much. You want the dough to grip the counter a bit.
Place them all in a pan. I prefer to use stoneware pans. I have several from the Pampered Chef that work great. If you are using a metal baking sheet, line with parchment first. Then place the rolls about 1/2 inch apart and brush with oil or melted butter. Keeping them close makes them push against each other when rising, instead of spreading and they will rise higher and become more 'pull apart' style. Cover again and let it rise. In the meantime, preheat the oven. You want them to rise and puff, but not double. They will rise in the oven as well. So they only need about 45 minutes. Keep an eye on them.
Just before popping in the oven, either brush again with melted butter or a bit of egg wash (an egg mixed with water). You can sprinkle a little wheat bran over them if you are using some whole wheat in the dough. Bake until golden brown, rotating the pan after the first 12 minutes, front to back. Depending on the size of your rolls about 18-20 minutes.